I don't know Newmarket, I am nearer Bury but I love Ely on the couple of times I have been there. there is a lovely little teashop up above the cathedral carpark, you go up the hill to the cathedral from the carpark and it is to your right, with a spectacular view. The Almoner's is another good tearoom with a lovely garden, it can be accessed from a walk alongside the cathedral or along past the walls. As you can see I find my way around by tearooms.
But Peacock's is just stupendous, as merlotgran mentions, Ely has a Riverside area which is almost like a separate village, with a walk and a pub and an art gallery and just off it, Peacock's.
Cambridge - well, just be prepared to walk a lot and explore. There is so much wonderful architecture that you will get a crick in your neck. Go down Bridge street to the river, turn right and walk past prezzo and a pub and you will find yourself on a boardwalk. Keep walking and you will be in the park, with a wonderful walk alongside the river, but never far away from the city. It's my son's back garden.
You might like the orchard tearooms, grantchester
http://www.orchard-grantchester.com/information/
There is the Fitzwilliam museum of course, but there is also the excellent Kettle's Yard
http://www.kettlesyard.co.uk/
full of music and crafts and art and yes, another teashop...
Fitzbillies is a famous cake shop near the Fitzwilliam. Wonderful chelsea buns.
I knew I should have had a pudding today. Why can't someone offer me a job of trialling teashops.