The most common medication for hypertension is amlodipine. (Countless AB members, including me, are on it). Information about it, including its possible side effects, can be found on the NHS website:
https://www.nhs.uk/medicines/amlodipine/
If my GP says that he's giving me a prescription, I just assume that I'll be able to pick it up from the on-site pharmacy (usually within 10 minutes, as long as its something that they keep in stock).
Hypertension is one of those things that certainly needs monitoring but that can easily be done at home. (I check my B.P., with two or three successive readings, every day). If it stays within a reasonable range, there's no point in troubling the GP any further (although my doctor does ask me to drop a list of my latest readings into the surgery about once a year anyway).
So I can't really see any problems in the approach adopted by your own surgery.