Is it possible to put a 700c wheel on to an old Raliegh road bike? I've been using a trainer during the winter months and went out on my bike the other day and half way round I got a flat!!!

I spoke to the bloke in a shop who said that using the bike on the trainer will wear the tyre out quicker. He mentioned getting an other tyre but said about the hassel of keep swapping them over. 

I have got an old Raliegh sprint with the old wheel size, I think it's 27 and 3/4 inch (not sure on that one!!) This however is too big for the trainer so my question is, Is it possible to get a new wheel which will be smaller than the one on the sprint to fit the bike and also the trainer. This will only be used on the trainer so it won't matter about the brakes.

Hope this makes sense!!
Sure.  So long as the hub is roughly the same width, and the chain is compatible with the sprockets, you should be able to put the slightly smaller 700c wheel on that bike.  As you note, the brakes will likely not reach, but that's not a problem on a trainer.

One potential problem is the rear derailer, but this wouldn't be as much trouble as going the other way, since your old derailer is likely not indexed.  The main problem is likely to be that the wider chain doesn't fit the sprockets well.