sarahrory This laptop is £420 and has a somewhat-faster graphics chip:
https://www.very.co.uk/lenovo-ideapad-slim-3-amd-ryzen-5-16gb-512gb-with-mouse-backpack-amp-earbuds/1601202402.prd
And this one is £500 and has a graphics chip that's somewhat faster than that:
https://www.currys.co.uk/products/acer-aspire-15-15.6-laptop-amd-ryzen-5-512-gb-ssd-grey-10284831.html
I don't see anything else that can match these two and that doesn't also have some other flaw (like a bad screen). If you don't like these for whatever reason, let me know why, and I'll list some options that fit your preferences and are a half-step down or so in terms of performance—there are plenty of laptops available in that tier. But these are the fastest.
I would say the first laptop is enough of an upgrade over the Omnibook, at a low-enough price difference, that it's an easy decision as long as you're comfortable going to £420. The second laptop is definitely a further upgrade, but if you don't need to play on the highest settings possible, it's not really necessary.
You wouldn't find a gaming laptop for £500 outside of an excellent Black Friday deal, at least not new, so the goal is to get fast-enough hardware to play at high-enough settings that you're happy while not losing out on another feature that matters.