I'm not a UK resident, but it was VERY obvious when I was there last summer that one just "shows up" for the trains -- they run very frequently, and reservations should not be needed, especially not on short hauls like London->Cambridge.

The trains I took to Cambridge (twice) were both nearly empty, and I just showed up at the station without checking timetables or anything.. 20minute wait, and along came a train.

Cheers