Helen; Bob pretty much nailed it in terms of the arrangements in the UK. The Luna and Simone is my choice for single/double/twin rooms when in London and they do have a family room(s) as well (I'm told). With a budget of GBP100-165 you shouldn't have a problem but probably closer to the higher number for 4 of you. Book early at the L&S though, they are popular for a reason.
Do some searching on this site and Trip Advisor Forums (if you're not familiar, they're pretty slick and their 'query' function is excellent) and you cut down the choices to a few in each city. With TAF, you can narrow down your searches to UK cities, never mind European 'districts'. Then ask opinions here, and you'll get no shortage of 'been there, done that' responses. I suspect you'll wind up in the middle of your 100-165 range but Google and see what's out there.
We stayed in a B&B in Bath last spring and it cost us GBP185/night for a double with a four poster bed. Lovely, classy, but wouldn't do it again -too many choices elsewhere and it was a special weekend ('nuff said).
My daughter worked in Oxford for 6 months after university and from listening to her, you won't find a shortage of places to bed down the family in one room. Can't say for Canterbury. You will, though, have a wonderful trip based on what I see from your itinerary.