I'm assuming that's with a change in Nottingham?
Lincoln to Notts is an East Midlands train, no cycle reservations, bikes normally by the toilet, sit next to it. Used it many times never a problem even when there's a couple more bikes than the permitted 2.
Notts - Derby is probably Cross Country, reservation are supposedly required, I've used it several times and never bothered, once or twice I had to wait for the next train but there's several an hour.
Depending on where in Derby you're heading - you might also consider staying on the first train and going to East Midlands Parkway, about 10 easy miles from Derby centre. I'm that side of town and it's easier for me than changing trains.
