Yes, assuming you can find the second hand parts. It could be worth considering it as a longer term winter project. Buy a frame, then list all the bits you want, and begin getting them from wherever you can. Join Freecycle and put a wanted ad on for old bike parts. You could be amazed at what can be offered. Car boot sales are a good source where you may even find a complete bike. Saw a guy at a car boot a few weeks with a very decrepid Raleigh Pursuit asking £250. Very rare and very collectable he told me. He seemed quite offended for some reason when I laughed in his face.
You say build a bike for around £200. If you can stretch to that have a look at the bike of one of the members on here, Angelfishsolo. Looking on the web they seem to get reasonable reviews, and there is one with different gear shifters within that budget. I'm sure Angelfish would be pleased to tell you more.
The computer on my Raleigh is indeed a
Halfords 6 function wireless. Don't forget there are two batteries, one in the main unit and one in the transmitter on the forks.