@lthomasdixon Sorry for the delay; I didn't have time to look for laptops until tonight. While £250 isn't much of a budget, I did find one decent used laptop for £260 on Amazon:
https://www.amazon.co.uk/ASUS-VivoBook-A556UQ-XO1470T-i3-7100U-Processor/dp/B07882J5Y2
(This is the link for the new version, but the used options are linked as well, and they're from Amazon Warehouse.) This is not a computer that would run all Sims 4 packs on the highest graphics settings, but it should handle high settings fine. Its main drawback is its 4 GB memory. However, it does have one free RAM slot for upgrades. Another 4 GB might cost £30. For example:
https://www.kingstonmemoryshop.co.uk/laptop/asus/a556uq-series/asus-a556uq-xo1470t-laptop
If you're willing to buy from Ebay U.K., there are a couple of even better options currently on sale. Both of these have a significantly stronger graphics card and stronger processors, as well as plenty of RAM. The second one is better than the first, although its hard drive is quite small. It's a solid state drive, which means it's much faster than the hard disk drive in the first model, but 256 GB might not be enough. It's plenty for Sims 4, web browsers, homework assignments, etc., but not for another game or a large music collection. On the other hand, external hard drives aren't that expensive either if your kid needs the extra storage.
https://www.ebay.co.uk/itm/ASUS-FX503V-Gaming-Laptop-GTX-1050-8GB-RAM-1TB-HDD/202756189560?hash=item2f3535ed78:g:jgkAAOSwgsFdUal-
https://www.ebay.co.uk/itm/Dell-XPS-15-9570-i5-16GB-256GB-SSD-FHD-laptop-with-GTX-1050/183918199338?hash=item2ad261022a:g:5wwAAOSwwtNdUbn3
I have no idea if you're comfortable buying from Ebay, but the improvement in hardware is significant enough to be worth mentioning.