Cambridge is within 2-3 hours of a number of airports:
This is closest of the London airports. Buses are available to Cambridge. There is also a train service from the airport station, which takes about 30 minutes to Cambridge station.
Luton Airport is about an hour away from Cambridge. There is a bus service from the airport to Cambridge. It is also possible to get from Luton to Cambridge by train via London, but this can take 2-3 hours depending on connections.
The journey from London City Airport to Cambridge takes around 2 hours. The airport is served by the Docklands Light Railway, which runs between City Airport and Bank Underground Station (a journey of approximately 22 minutes) for Underground connections to Kings Cross Rail Station, from where there are trains to Cambridge. There is also a DLR service to Canning Town (journey time 8 minutes) from where you can take the Underground into central London.
Heathrow airport is served by regular coaches to Cambridge. The scheduled journey time is 2-3 hours. Because of heavy traffic around London, the actual departure times of these buses are sometimes very different to the times listed in the timetable. You just have to locate the correct bus stop and then wait for one to come. Alternatively you can take the underground from Heathrow to central London, and then a train to Cambridge. A third option is the Heathrow Express to Paddington Rail Station, followed by the underground to Kings Cross and then a train to Cambridge. All these options take about the same time to get to Cambridge.
Gatwick airport is also served by regular coaches to Cambridge. The scheduled journey time is at least 4 hours. Because of heavy traffic around London, the actual departure times of these buses are sometimes very different to the times listed in the timetable. You just have to locate the correct bus stop and then wait for one to come. There is also a direct train service from Gatwick to St Pancras station, London. From there you can walk across the road to King's Cross station where there are regular trains to Cambridge. Alternatively, you can take the Gatwick Express to London Victoria Rail Station, followed by the underground to Kings Cross, and then a train to Cambridge. Using the train will typically take about 3 hours.
The simplest way to get from Birmingham airport to Cambridge is probably to take a coach. Alternatively, you can take a train from Birminham International Rail Station (which is connected to the airport by the "Air-Rail Link" monorail system). Journey time is typically at least 3 hours in both cases.