I'd choose the AM4 build if you only game, but the Intel build if you do creative/productivity stuff that needs more CPU power.
AM4 MATX build:
AMD Ryzen 5 5600 Six Core 4.4GHz (Socket AM4) Processor - Retail - £149.99
Gigabyte B550M AORUS ELITE (AMD AM4) B550 Micro-ATX Motherboard - £99.95
Team Group Vulcan Z T-Force 32GB (2x16GB) DDR4 PC4-25600C16 3200MHz Dual Channel Kit - Grey - £79.99
PowerColor Radeon RX 6700 XT Fighter 12GB GDDR6 PCI-Express Graphics Card - £389.98
Lian Li Lancool 205M Mesh Micro-ATX PC Case Black - £69.95
Super Flower Leadex III 750W 80 PLUS Gold Modular Power Supply - Black - £99.95
WD Black SN770 1TB SSD M.2 2280 NVME PCI-E Gen4 Solid State Drive (WDS100T3X0E) - £74.99
WD Blue SN570 500GB SSD NVME M.2 2280 PCIe Gen3 Solid State Drive (WDS500G3B0C) - £42.95
Grand Total: £1,020.05 (includes delivery charge, £12.30)
Intel MATX build:
Same case & PSU.
The i5-13500 is cheaper than £260, so should be near budget after the swap.
* Intel Core i5-12600K 3.70GHz (Alder Lake) Socket LGA1700 Processor - OEM - £259.99 (* SWAP for i5-13500)
Gigabyte B760M DS3H DDR4 (LGA 1700) DDR4 Micro-ATX Motherboard - £114.99
Team Group Vulcan Z T-Force 32GB (2x16GB) DDR4 PC4-25600C16 3200MHz Dual Channel Kit - Grey - £79.99
Gigabyte Radeon RX 6650 XT Eagle 8GB GDDR6 PCI-Express Graphics Card - £319.99
WD Black SN770 1TB SSD M.2 2280 NVME PCI-E Gen4 Solid State Drive (WDS100T3X0E) - £74.98
Grand Total: £1,032.14 (includes delivery charge, £12.30)