You could and maybe should do both. I learned the craft on the Swale at Richmond which had all sorts in it (Grayling, Trout, Dace, Chub and Barbel). 70% of the time I course fished and 30% flyfished. You could certainly fly fish on the nidd, ure, wharfe as well as the Swale. The Swale is a really lovely river, just brown trout above the falls at Richmond but increasingly course fish as you move down river.
Fly fishing doesn't need to be expensive. You could get all you need for £200 to get you started. A 5 weight rod and floating line is all you need to start and then add to it as you go. A couple of hours with a casting coach would be good money spent to get you up and running. Happy to go into as much detail as you want mate but there are days that you just want to stay light on the river and cover a lot of water and fly fishing is ideal for that. I don't course fish now living in Scotland but I wouldnt imagine it would be any more costly to get started again with a general purpose rod and fixed spool reel. I went Barbel fishing on the Wye and fully expected to fall in love with Course fishing all over again. I had a great day but still wouldnt swap it for casting the fly.
Fly fishing also opens up a brand new hobby - fly tying, but enough of that here.