They do, but services did change frequently in the early days of deregulation. I've only ever been to Oakham once and I didn't go by bus, but my memory is that the only bus I saw in Oakham was a Transit in Cambus two tone blue. Did Cambus have a Leicestershire contract for Stamford - Oakham, and their runs between Peterborough and Stamford were - at least in part - vehicle positioning for this? (This must have been before 1989, when the Peterborough end of Cambus became Viscount and the buses turned yellow.)
A year or two later I was bus bashing in the Peterborough area. That day included bygone names like Elsey's (Gosberton), W H Fowler (Holbeach Drove), and J R Morley (Whittlesey), and whoever was operating March - Wisbech at that time. Might that have been Emblings Coaches (Guyhirn)? Coming back to the point, I saw a vehicle from Bland (Stamford) at Peterborough Bus Station and it was heading to Stamford for a connection to Oakham. Who operated the connecting service by now I'm not sure.