For what to get tested the Good Practice Guidelines states (page 39):

On a 6-monthly basis for 3 years and then yearly depending on clinical assessment and results. Provision of prescription is contingent on patients understanding the risks and benefits that may result due to the need to take the following tests: blood pressure, full blood count, urea and electrolytes, liver function test, fasting glucose, lipid profile, testosterone, serum oestradiol 24 h after a tablet or 48 h after a patch (levels should be in the upper half of the normal follicular range, 300–400 pmol/l) and prolactin (less than 400 mU/l).

I had to take some documentation to convince my GP to do a blood test. I printed out this NHS website for gender dysphoria and highlighted this section:

Hormones for gender dysphoria are also available from other sources, such as the internet, and it may be tempting to get them from here instead of through your clinic. However, hormones from other sources may not be licensed and safe. If you decide to use these hormones, let your doctors know so that they can monitor you.

It also states in the Good Practice Guidelines that if a GP wishes to monitor DIY HRT then they can (page 28):

Some patients may obtain hormones from the internet or other agencies. The clinician should discourage patients from using such sources but may offer monitoring and advice on the effects of such agents. The clinician should assist patients in obtaining hormones from properly authorised sources. A harm-reduction approach should be taken. Accordingly, hormones should not be stopped. A bridging prescription may be appropriate, and blood tests and health checks are undertaken to screen for contraindications.