I agree with Lin. Premier Inns are very good and very clean and comfortable hotels that are very reasonably priced, and you can find them in any neighborhood in London and also in Cambridge. They have family rooms available and will provide a "travel cot" for babies (similar to a pack-and-play). Breakfast is available but is not included in the price, although it's usually pretty reasonable.
Here's the website for their main page, and you can search for the neighborhood you want in London or in Cambridge. I would imagine that the Cambridge locations will be cheaper than the London locations, so you might want to check there first, but definitely compare prices. Also, keep in mind that the earlier you book at Premier Inn, the lower the price. They have a number of price options:
- One that is completely refundable, which will be the highest priced
- A hybrid where it's refundable up to 25 days ahead of time, but you pay up front
- The cheapest option, which is non-refundable and you pay up front
This PI site lists the Cambridge hotels. https://www.premierinn.com/gb/en/hotels/england/cambridgeshire/cambridge.html
There too many to list the London, but you can check those out on your own: https://www.premierinn.com/gb/en/why/family.html