Thanks all!
It looks like the AM4 socket is definitely the way to go. Just need to make a decision on which CPU and which GPU. With the 3950X and the Radeon VII it's a little under £2,100, or with a 2080 Ti it would be closer to £2,700.
PCPartPicker Part List
CPU: AMD Ryzen 9 3950X 3.5 GHz 16-Core Processor (£698.87 @ CCL Computers) CPU Cooler: Noctua NH-D15 CHROMAX.BLACK 82.52 CFM CPU Cooler (£89.99 @ Amazon UK) Motherboard: Asus PRIME X570-P ATX AM4 Motherboard (£153.99 @ AWD-IT) Memory: Corsair Vengeance LPX 64 GB (4 x 16 GB) DDR4-3200 Memory (£281.51 @ Overclockers.co.uk) Storage: Sabrent Rocket 1 TB M.2-2280 NVME Solid State Drive (£119.99 @ Amazon UK) Video Card: XFX Radeon VII 16 GB Video Card (£512.00 @ Amazon UK) Case: Fractal Design Define S ATX Mid Tower Case (£67.97 @ Amazon UK) Power Supply: Corsair HX Platinum 850 W 80+ Platinum Certified Fully Modular ATX Power Supply (£147.37 @ More Computers) Total: £2071.69
That leaves two last questions, how likely is it that AMD will keep producing new CPUs for the AM4 socket for a while to come? And what's the best way of selling a secondhand PC? I could take most of the drives out and sell it as a working machine, take it apart, reuse the case and PSU and sell the rest as parts, or I could give it to my wife to game on...