Routine GPs appointments can generally be gotten within a week, and emergency[1] appointments (where we wait until the GP is done at the end of the surgery, and are then seen) if the triage nurse determines you need to be seen sooner can be gotten on the day. You also have the option of speaking to your local pharmacist, many of whom can prescribe medication for certain conditions if appropriate, or otherwise point you at suitable over-the-counter primary care.

A number of places also have walk-in centres for minor ailments where you don't need an appointment, and they generally have long opening hours (8am-10pm, 365 days a year for my local one). Urgent dental care is also available 24/7 in many places.

You may have to wait longer for non-emergency / non-priority / time-insensitive treatments - my routine MRIs have a lead time of about 8 weeks, and a specialist referral can be something in the region of 6-12 weeks (sometimes longer) - but I think that's a fair trade off. You do of course also have the option of paying for private care as well if you don't want to wait (often NHS doctors / semi-retired doctors working evenings in my limited experience).

[1] bit of a misnomer really - urgent but not critical is more accurate