PC Master Race

600 British pounds is ~771 USD If only new parts were picked, following the https://www.logicalincrements.com/ guide good tier

GPU: RX 6600 $200

CPU: i5 12400F $148

HSF: Peerless $35

Mobo: ASRock B660 Steel $120

RAM: 8GB DDR4 $24

Hard Drive: 2TB $52

M.2 SSD: 512GB $42

PSU: SeaSonic 500W (B) $60

Case: Pure Base 500 $80

Personally, I would change this to get rid of the Hard drive to free up ~$50, putting the money into a higher capacity M.2 if needed

I think you would probably be looking at 1080P gaming, and I don't think this built will be super upgradeable (DDR4 RAM, Intel CPU) but I think you would be able to play games at a reasonable framerate

definitely used at that budget, local if you can, go for older gen cpu to leave room for gpu, like 8th gen intel or ryzen 1600, with a 2070 or even better

I'd get a Steam Deck with that money.

You will have to look at the second hand market for sure, but you can probably get a decent 1080p 80-100fps medium/high build.

Something like (or equivalent):

Ryzen 5 3600

RX 6600

16GB DDR4

1TB SSD (probably not NVMe)

Whatever case

Whatever monitor, even if only 60Hz, for now

Whatever MoBo

Tier A PSU from the Cultists Network PSU tier list

You can most likely get everything used if you buy from a second hand shop that has a form of buyer protection (eBay and similar) and thoroughly check and clean and test everything you buy

New: Motherboard, Heatsink, and PSU Used: CPU, RAM, GPU, and Case

The majority of components you find on the used market are pulled out of working systems and will be just fine for years to come. I only recommend buying those specific items new so you are guaranteed to get all the accessories that come with them.

At that budget you are limited to a second hand gpu, possibly something like a 3070 if you're lucky. Maybe get like a 6700XT. Ryzen 5 3600X, 16GB DDR4 and a simple motherboard.

Hmm at this budget you're likely shopping for deals. Mb cpu combos. Basically stuff bundled together. That's likely to dictate what parts you get to buy.