I found the NHS more hands off until required than my US doctors, but not in a bad way. A lot of it will depend on area. I live in London.
No yearly physical. I am invited for a women's health check every 3 years. If I don't make the appointment I normally get a follow up letter. The health check is with the nurse and is a fairly quick heart rate, blood pressure, any problems?, what do you eat, how much do you drink, and pap smear. The results are mailed within a couple weeks, but I'm told I'd receive a phone call before then if something was wrong.
The sexual health clinic is a weird one for me. They deal with my birth control, not my GP, so I see them more frequently. I have mixed feelings about it, mostly because the clinic is seemingly aimed at younger patients and they like to do drops-in on a first come, first served basis. I insist on appointments. The sexual health clinic is also much further away which is very annoying. The doctors there are wonderful, I can't complain about them.
I did once find a lump in my breast. The NHS moves quickly! I was whisked straight for 2 hospital appointments even though I was in my early 30s with no family history of problems. I don't remember all the details, but the referrals were immediate and I'm sure I was at the hospital in less than 48 hours after casually mentioning the lump. It was deemed to be nothing, but I still had a follow up appointment and a 1 to 1 with a doctor who went through how to check, what to expect, and what the lump was (it was some type of hormonal bump for me). The doctors were wonderful and while I felt guilty in wasting their time they were very through and my mistake wasn't a problem to them at all. I wouldn't hesitate to report another potential issue, even if it turned out to be a non-issue.
My NHS dentist doesn't remind me to have check-ups like my dentist in the US did. I find that difficult. I can make appointments however frequently I want though and I like my dentist. I disliked my previous 2 dentists. I go about once a year. I get the teeth check, gum check, questions about what I eat and drink, and if there's any pain. He does clean my teeth every time which is part of the reason I haven't changed dentists even though I now live an hour away. I'm offered x-rays every 2 years, though I normally manage to put them off until every 3 years.