Not only are trains sleeping on Christmas Day, they sleep on Boxing Day too. There is often one National Express bus between Heathrow and Victoria Coach Station but don't expect any others to run on Christmas Day and only some on Boxing Day.
I don't know about local buses in Cambridge. I visit there frequently but I've never researched their buses over the week between Christmas and New Year.
The changes in effect last year are at:-
https://www.cambridge-news.co.uk/news/stagecoach-christmas-bus-timetable-changes-15546162
Now that I've looked at that, I see that last year National Express ran a few services from Cambridge direct to Victoria Coach Station in London for the first time. I have absolutely no idea if they will repeat the experiment this year. These sort of information stories usually come out about a fortnight before Christmas - maybe you will be lucky and information might be earlier this year. Probably not because public transport relies on staff working overtime and volunteering to work on the holiday - and the union has to agree the pay scale for those special days, and that doesn't usually finish until the roster is posted.
You will see that last year there were no local services in Cambridge on either Christmas Day or Boxing Day.
The week between Christmas and New Year is usually the time of year when the railways do their most major projects, often closing entire London terminus stations and putting in extensive diversions. That won't normally be posted any earlier than 12 weeks out, and maybe not that early.
Two things that I would not normally advise are renting a car to go to Cambridge as parking is difficult and expensive, and the same in London, but in some towns and cities parking restrictions are lifted on Christmas Day - but not usually Boxing Day - so that might work if you can get it early enough and return it late enough to work for you; or having a taxi. They will probably be working to some degree but the prices will likely be through the roof and Cambridge is not exactly close to London.
Or find a nice place to stay and enjoy Cambridge for a day or two over the Christmas season.
Good luck - this isn't an easy nut to crack and still keep a hold on your pursestrings.
