The Ryzen 1600 is actually a 6-core/12-thread processor, better than both the 4-core/8-thread 2400G and the 2-core/4-thread G5400, and should remain relevant longer. It lacks integrated graphics though, so it requires a dedicated graphics card.
However, I would definitely recommend a dedicated graphics card anyway for a gaming build, since a Radeon RX 570 offers 3-4 times the gaming performance of the 2400G's integrated graphics.
I did cut back to 8GB of RAM, but a dedicated graphics card with its own 4GB of video RAM should lighten the load on system RAM compared to when running integrated graphics.
And since it's first-gen Ryzen, I was able to go with an X370 motherboard with 4 RAM slots, which could allow for an additional kit of RAM to be added later.
The remaining parts are similar to what's in the other builds, since they are some of the least expensive options to help keep the budget at least somewhat in check. In my opinion, it's probably worth paying that bit extra so that you don't have to choose between having a capable CPU with weak graphics hardware, or a capable graphics card with a weak CPU.
And if you could spend an additional £30 beyond that, it would probably be best used moving up to a 16GB (2x8GB) kit of DDR4-3000 RAM like in the first build posted by Phaaze, as you will likely want it eventually.