In our county, primary music lessons are quite heavily subsidised, so you pay £40 a term for 1 lesson a week - lessons are usually about 20 mins, and may or not be shared depending on if anyone else plays that instrument at a similar level. So, my son got individual lessons because he was the only sax player in the school.
You have to provide your own instrument, although there are a few ones which can be borrowed.
Recorder clubs at school (lunchtime) are free, run by several teachers just out of goodwill and to build up a decent recorder ensemble!
At secondary school, we pay £120 a term for 10 lessons a term, each one 30 mins of one-to-one tuition. This seems to be the going rate for all the freelance music teachers. Year Sevens get 10% discount.
It certainly is expensive, but worth it if your child is musical. My 3 are only allowed one expensive paid-for activity like this each, the first 2 do music the 3rd hasn't decided yet.
My eldest is desperate to do a second instrument - she does flute now and wants to do keyboard - and I would love her to because she is so musical. She is taking grade 5 flute in Nov, only 20 months after first picking up the flute. But for now, she knows we can't afford a second instrument so is teaching herself!