This depends mainly on 2 things - the species, and how successful the nest is. Short-lived nests such as Bombus pratorum can be over in just a month or so, whereas a Bombus terrestris could last from early spring to October. So first identify your bumblebee, and remember that, sadly, not all nests go through the whole lifecycle.
Yes you can if you build a suitable nest box, of course they will not survive the winter. It is not easy though, and I have never got past the stage where the queen has made her honeypot. Designs for nest boxes can be found in Sladen, Alford and Pry-Jones's books or on the nest box plans page.
Really the best answer to this question is do nothing and leave the bumblebees to nest in peace! You can, of course, look forward to enjoying watching the bees come and go. The farmers who grow tomatoes under glass pay a fortune for bumblebee nests; yours is free, and at the end of the nesting period you will have a bumblebee nest to look at and examine. You can try following the bees to see if they all visit similar shape and colour flowers. In fact you should consider yourself quite lucky.
However I realize that you may be looking at these pages because you are worried about where the nest is located and what might happen. You may not feel lucky after all. Firstly I must reassure you that bumblebee nests are not like honey bee hives, they last only a few months, or even a few weeks, and are usually small enough to hold, and bumblebees are not as ferocious as wasps. The bees are fairly placid and are unlikely to sting unless they feel their nest is threatened. So if the nest is under the house or shed it is best just to leave it. Bumblebees do not damage brickwork or wood.
Some of the bumblebees which make smaller nests, Bombus pratorum and B. hortorum do sometimes chose strange places to nest. Their nests are small and in the case of B. pratorum are usually very short lived. So again, I would say leave the nest alone. If this is impossible, for example if the nest is under your rotary lawnmower that you forgot to clean last year, or in the pocket of your gardening jacket that you left in the shed at the end of last summer. You may feel that you have no choice.
Well, you can try to move the nest to somewhere more suitable, or even better, remove the lawnmower. Read my section on stings before attempting to move a nest. Firstly you need a container big enough to hold the nest. Anything fairly strong will do, a sandwich box, a small biscuit tin, an old teapot. Or you could make a nest box, almost any weatherproof container will do as long as you make a hole big enough for the bees to get in and out. If you can get hold of the book by Sladen, Alford or Prys-Jones, they have many designs for artificial nests, or have a look at nest box plans. If the nest is to be placed outside then the container must be weatherproof, but if you found the nest in a shed then it is best to leave it in the shed, it won't cause you any bother as you come and go, and the bees obviously prefer it.
Next you should try to move the nest either late in the afternoon or early evening when it is both cooler and darker. Most of the bees will be in the nest, and as it is cooler they will be less active. Get someone to help you and work quickly. You can gather up as much of the nest material, grass moss or whatever; alternatively you can provide clean nesting material that will be free of parasites, dried moss, cut up pieces of dried grass, felt etc. If you are leaving the nest in more or less the same position but removing the lawnmower or jacket, then this is all you really need to do. If you are moving the nest to a different location then you must be prepared to catch any late returning workers and place them in the new nest. The workers will return that night or next morning once it has warmed up. You really do need to collect these workers, the number of food gathering workers in these nests is quite low and the loss of only a few can mean the nest will not have enough food to survive. The nest should be placed in a sheltered south-facing location.
If the bees are the underground nesting kind things are more difficult. The tunnel to a Bombus terrestris nest can be two metres long. This calls for quite a lot of digging, and really they are best left alone. However you can try to "move" the exit/entrance of some nests by adding a bit of hosepipe leading from the real exit/entrance. Do not add a long piece - less than a metre - is about the limit, and do this at night time.
The quick answer is put it out and leave it to weather. If there are bumblebees about then you could try to identify them to find out if they are ground, surface or above ground nesters. Although most species have a preference, they do often occupy a nest that is not in their preferred location. Then if the nestbox does not have nesting material you can take your time to make the inside of the box as bumblebee friendly as you can, see the nestbox page for more details. Next look around for a good site. And having found the site take your time to place the box well. And then go away and leave it. In early spring have a look inside the box. If it shows signs of occupancy by roosting birds or small mammals, then this is a good sign. Leave any nesting material and droppings, bumblebee queens seem to like the lived-in smell and look. Put in a little more nesting material if you think it needs it. However if the nesting material is damp and the box has no bird/mammal sign, but has slug/snail/worm signs, then it is not well sited. So you must take it out. Dry it thoroughly - don't clean it up too much and don't use any cleaners. Then find another spot for it.
In the UK the biggest nests are built by Bombus terrestris and according to Sladen in his book even the biggest nests never contain more than a few ounces. In one B. lapidarius nest that he reared in a nest box he took over four ounces of honey and the nest still survived. So yes, in theory we can get honey from bumblebees as it is made by the same method as hive bee honey, but the quantity is so small that it would never be commercially viable to do so. Also, I believes, that it is thinner and more watery than honey bee honey, so ferments more easily. One average sized bumblebee making about ten foraging flights a day would bring back enough nectar to make about 3 ml of honey. But to recover the honey you would have to break into the nest, destroy the honey pot during extraction and probably destroy eggs and young in the process. Bumblebees nests are not neat organized affairs like honey bees nests, they are rather untidy and disorganized.
Honey is really just concentrated nectar, and nectar is mainly a mix of different sugars that are secreted from flowers into their nectaries. The bees suck up the nectar using their tongues. The tongue is long and feathery at the end. It is contained in a sheath formed by a pair each of palps and maxilla (these are just mouthparts). Together the palps and maxillae act a little like a straw, so the bee sucks the nectar up this and into her honeystomach. The honeystomach is just a storage bag, and when she gets back to the nest the bee empties the honeystomach into a honeypot. So all the nectar that has been sucked up through her mouthparts has to pass back up through her mouth again. There may also be a few pollen grains mixed in with the nectar too. The temperature of the bee's body is almost the same as a human's so while she has been flying around some water in the nectar will have evaporated, and also inside the nest it is quite warm, so again more water will evaporate. After a while, once more water has evaporated from the honeypot the sugars are more concentrated and the contents can be called honey. It is said that honeybee honey is more concentrated than bumblebee honey, and because of this honeybee honey can keep longer as the high sugar content preserves it better.
To make a one pound jar of honey the honeybees must visit and suck nectar from about 3,000,000 flowers of red clover or 2,000,000 flowers of vetch. Think of all that work next time you spread it on your toast!
There are three types of honey, flower, which I have described above, forest and leaf honey. The source of these other two is very interesting. Quite simply they are made from the excrement of aphids and scale insects. These insects suck the sap of plants, but the sap contains much more sugar than they need, so they excrete the extra sugar. This is what we call honeydew. Now these insects can exist in huge numbers so the honeydew can be splattered over the leaves and pine needles of plants (it is also this which car drivers find gumming up their windows in summer) the bees collect this lovely sugary liquid in the way described above (ants also collect it, and even "milk" aphids for it, and so can be considered the very first farmers in the world, see this page for a photograph of this in action) and it is treated just like nectar. In the bible the manna from heaven that rained down on the Jews as they passed through the desert is thought to have been honeydew from a type of scale insect commonly found on tamarisk.
I get quite a few emails about this. While most of you are content to leave the established nest until it comes to a natural end, many do not want another nest next year. The solution is fairly simple. You do not need to block up the air brick, and anyway it is there for a reason. All you need to do is cover it with either metal or plastic mesh small enough to prevent a bumblebee queen entering. Bumblebees have weak jaws - only strong enough for forming wax - so the mesh does not have to be heavy duty stuff. Just fix it there during the winter, and no bumblebee queens will be able to enter next spring.