Welcome to TP, James. For bird photography, you typically want the longest focal length you can manage. A 2x extender will turn your zoom into a 140-400mm f/5.6, which is the same as the 100-400 at the long end (i.e. 400mm f/5.6). The 100-400 has IS (you don't say whether or not your 70-200 has), and it will give you slightly better image quality than a 70-200 plus 2x, but to be honest I don't think those advantages are going to be worth paying best part of £1000 for. Given that you already have the 70-200, a 2x Extender (~£200) is going to be the most economical way of getting to 400mm.
Beyond that, the prices start to look scary. The 300mm f/2.8 IS is a lovely lens that takes a 2x Extender very well to give you 600mm f/5.6, but it's around £3000. The 500mm f/4 IS is another birders' favourite but it's around £4000. But if you try to economise with a 300mm f/4 IS and 2x, or the 400mm f/5.6 and 1.4x (each combo is about £1000), your fastest aperture is only f/8 and that's not going to be good enough (especially without IS) except in bright light.