Seems to me a big decision point is whether you really need a new rear wheel or not. If the wheel is straight and rolls well, a little rust probably won't mean anything. a 6 speed is most likely a freewheel. You should be able to fit a 7 speed freewheel on there as the spacing is normally the same (check if it's 126 mm between the dropouts on the frame). 6 speed and 7 speed freewheels are readily available in the US. I would think they would be in UK as well.
A shifter is tougher. If you want indexed shifting, probably tough to find anything 6 speed, but I think you can still get 7's. Just depends on what type you want (frame mount, stem mount, etc.) But if the current is friction, you can probably clean it up and get it functional pretty easily unless it is bent or missing parts.
For the derailleur, you would just match the type that is on there. If the current one has a built in hanger, get the same.
I would say new brake cables, pads, tubes and tyres are a given. This is safety stuff so get something decent.
Changing bars is either simple or more complex depending on what you're changing to/from. Would need more detail.
Unless you have to change wheels or really change the whole bar/lever set up, you can probably get in in good shape pretty cheap. But +1 on amillerinmaine's point. Really go through and decide what you're going to do before you start buying stuff. If you just buy things and throw them on without a plan, you can end up spending what you would ahve paid for a new bike pretty quick.