Public transport in Cambridge is awful. Your best option is to probably look at the guided busway. So moving out of Cambridge you would have Milton, Histon and Impington, Oakington, Northstowe, Swavesey, Fen Drayton and St Ives. Prices of homes in these villages will decrease as you move out of Cambridge but are still expensive villages. St Ives is the biggest of those and is a small town with a small shopping area, a couple of primary schools and a secondary school. The guided busway can be really busy and if you’re in the closer villages you might not get a seat at peak times.
You can catch the train but you would have to catch the guided bus from the station to Addenbrookes, or cycle. Cambridge south station is due to open in 2025 and that will be near Addenbrookes. However, when they opened Cambridge North they didn’t get the timetable right and you have to catch a train into Cambridge then back out to Cambridge North. Cambridge South might be the same. Villages that you can catch the train from are Milton (Cambridge North) but you’re better sticking to the guided bus if you choose Milton. Also Waterbeach to the north or Shelford, Whittlesford, Great Chesterford, Shepreth, Meldreth and Foxton to the south. Again all these villages are expensive because of their train links to Cambridge and London. Trains are roughly every 40-60 mins and don’t run 24 hr a day. You could choose Manea or March in the Fens, these areas are cheaper but trains are a lot more infrequent and Manea is a request stop. You can also consider Peterborough by train, much cheaper than Cambridge but the train takes ages.
If you can cycle, you can probably consider some of the villages south of Cambridge, Trumpington, Great Shelford, Fulbourn, Babraham and cycle to Addenbrookes. Regular buses generally take forever and get stuck in the awful Cambridge traffic especially at rush hour. There are bus lanes but not enough to actually allow buses to get anywhere. The guided busway is a separate lane and those buses can travel relatively easily. There is also a cycle Lane along the guided busway.
Addenbrookes is also close to a number of schools and colleges. Most students in Cambridgeshire will end up at Hills Rd sixth form or Long Rd, which are right by Addenbrookes. So if your shift is starting at school time, the guided bus, buses and trains will all be full. If you cycle, there will be teenagers darting out everywhere.