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You definitely don't need a dedicated graphics card. I've run it on a couple of 5+ year old second hand ThinkPads with Intel integrated graphics absolutely fine.
Edit: just checked and I currently have an X260 with an i3-6100U and Intel HD520 graphics, 1080p screen. Probably from CEX. Graphics almost indistinguishable from when I used it on my full-on desktop.
I do remember that Zwift have dropped support for older Intel graphics though, so you might want a bit of future-proofing.
Edit- this is all about Zwift, don't know about TR.
The consensus seems to be that I can’t get away with spending as little as £300 for Zwift.
What is the cheapest laptop that will run it comfortably?