You could cycle pretty much on any bike that fits you.... course, some are more comfortable and suitable than others!
In these days of index shifting, carbon fibre frames, 30 speeds, STI/Ergo shifters, etc, it's easy to forget that people back in the early 20th century covered huge distances on bikes we wouldn't even consider now. That's not to say I'd want to do the same (happy with my more modern touring bike with disc brakes and index shifting, thanks) but isn't touring more about the fun and the journey rather than the equipment?
If you don't mind going slower than a road bike, or a purpose built tourer, converting an old mtb with good quality components can be very easy and cheap. Slap on a couple of thinner higher pressure tyres and tubes, a bar bag, rack (certainly older mtbs usually have rack mounts) and panniers and you can be good to go. Add in some stem/quill changes and bar ends or a different handlebar setup (if you stick to straights or butterflys you can use the same gear shifters) and it's even better. If you feel the need, add some mud guards too (though it doesn't rain in Britain, right?) Even the gearing doesn't need to be changed (trust me, on a tour usually you're looking for lower gears rather than higher ones!).
Sounds like you've come to this conclusion.
I should add I'm not against someone buying a purpose tourer, just that if you're only going to tour once or just want to try it out, usually an mtb is a cheaper option -as well as you can work out what you do like.