Is it possible to put a 700c wheel on to an old Raliegh road bike?
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.