@bondzie_j There are a few different categories of laptop to consider. Sims 4 (and almost all modern games) run much better on at least a mid-range dedicated graphics card, which will be significantly faster than the best integrated cards. In between, there are some low-end dedicated cards. While other specs matter, they're a lot simpler. You need at least 8 GB RAM and enough storage—256 GB is doable if you're only installing Sims 4 and small apps like browsers, but you'd want more if you were planning to play other large games. Almost any decent dedicated graphics card will come with a processor that's good enough, and the best integrated cards are integrated into at least decent processors. So it's usually easiest to look at the card first and then check the other specs afterwards.
Within your preferred budget, you won't be able to find a new laptop with a dedicated card. The best available integrated card in that price range is the AMD R8, which would handle Sims 4, all packs included, on at least medium settings. If you had fewer expansion packs or didn't mind your in-game framerates being a bit lower, you might be able to run the game on high settings, but it would be easy enough to experiment.
I looked at a number of U.K. sites, and laptops with an R8, 8 GB RAM, and at least 256 GB storage start at £350. The equivalent Amazon models, or at least the ones from Amazon itself rather than a third-party vendor are more expensive—the cheapest I saw was £400. (Currys was even more expensive).
https://www.amazon.co.uk/ASUS-VivoBook-X512DA-R5-3500-Windows/dp/B07SHT8C3D/
For comparison, these are equivalent models from other sites. The first group has a 256 GB solid state drive.
https://www.laptopsdirect.co.uk/lenovo-v155-15api-amd-ryzen-5-3500u-8gb-256gb-ssd-15.6-inch-radeon-vega-8-w-81v50008uk/version.asp
https://www.laptopsdirect.co.uk/hp-255-amd-ryzen-5-8gb-256gb-ssd-15.6-inch-windows-10-home-laptop-8ab98es/version.asp
https://www.laptopoutlet.co.uk/hp-255-g7-15-6-full-hd-laptop-amd-ryzen-5-2500u-8gb-ram-256gb-ssd-windows-10-8ab98es-abu.html?___SID=U
(the second and third are the same model)
https://www.argos.co.uk/product/3068984
This one a 1 TB mechanical drive. Windows will run faster, and apps will load faster, on an SSD, although most apps, Sims 4 included, will run just fine on any drive once loaded. But if you need more storage, the larger drive might be a better option.
https://www.laptopsdirect.co.uk/acer-aspire-3-amd-ryzen-5-2500u-8gb-1tb-amd-radeon-vega-8-15.6-inch-windows-nx.gy9ek.009/version.asp
Laptops with a mid-range dedicated card usually start over £700, although these two are £600:
https://www.currys.co.uk/gbuk/computing/laptops/laptops/asus-fx505dy-15-6-gaming-laptop-amd-ryzen-5-rx-560x-256-gb-ssd-10194549-pdt.html
https://www.box.co.uk/HP-Pavilion-Gaming-15-ec0001na_2788805.html
There aren't a lof of options with low-end dedicated cards, at least not for cheaper than £600. This is the only good one I found that's significantly cheaper:
https://www.laptopsdirect.co.uk/lenovo-v330-15ikb-core-i5-8250u-8gb-256gb-ssd-15.6-inch-windows-10-home-la-81ax011muk/version.asp
Its performance would be a bit better than the laptops with an AMD R8, but maybe not £100 better. Still, it's a viable option.
If you have other questions, or want to see different options, feel free to ask.