I work in sexual health here. Your local NHS sexual health clinic will almost certainly have at home test kits that will test for chlamydia, gonorrhoea, hepatitis, syphilis and HIV. There is a bit of variability in how accessible the services are depending on where in the UK you live (and what specific tests you are offered, for example, if you aren't a gay man, you may not be offered a syphilis test), but I would check at SH:24, which is an NHS service. It should give you an idea of what you can order.
https://sh24.org.uk/
These tests are reliable and they're free. It's how most of our patients who test online test.
If you don't want to go via the NHS - and I should add that there is no reason not to, your sexual health record is never linked back to any other record, like your GP or another service you might use - then I would order a test kit from one of the well-known pharmacy chains, Boots, Supedrug, etc.
If you want something like herpes or HPV tested, I would speak to someone at your local sexual health clinic, as these aren't routinely done and you can't do them yourself at home. But if you have symptoms and are concerned, they can bring you in and test you.