A 580 won't even give 60 FPS maxed out at 1080p in most newer games, don't know why it's being suggested. Freesync or not, especially over a GTX 970. It's just burning money.
Buy nothing unless you're able to get a Vega 56 cheap or 1070/Ti.
Still, OP's main issue he wants to fix is screen tearing and surely Freesync would fix that, and would still look smoother compared to using a GPU without Freesync even with higher FPS. The other alternative option is to maybe sell his current monitor and combine that with his current £350 budget and stretch to a G-sync monitor. Saying that, OcUK seems to have this on offer which is right in his budget (£349.99 as of this time), but then again, he hasn't mentioned what size his current monitor is

But yes, I do agree if he has to upgrade the GPU, then Vega 56/1070Ti seems to be the sensible option.