To my mind this is the ultimate in child friendly Cumbrian bike rides. It ticks all my boxes - off-road, flat (amazing!), scenic, and somewhere fab to visit at the end of it.
Annoyingly I have no photos from the day we did it, so I am borrowing this one from Visit Cumbria http://www.visitcumbria.com/kes/keswick-railway-footpath.htm hope they don't mind.
So the way to do it is to park at Threlkeld, and don't fret about this, there is an old school or community hall, can't remember which, but it's well signposted with plenty of car parking and handily some toilets. Leave the car there then head off on bike or foot (plenty do) through the village towards the Keswick turn off of the A66. Just to the right of the big turn off is a path that goes down to the track.
The track is actually part of the old Cockermouth, Keswick and Penrith railway which no longer exists. It has been made into a terrific bike and walking trail with some gorgeous old railway bridges to cross. It's about four miles in length so it will test the little ones but there are lots of places along the River Greta where you can stop and rest and throw in stones. You'll never be alone on this ride, it was suprisingly busy with other riders and walkers when we went, but the atmosphere was really nice and cheery. The great thing about it is that it comes into Keswick right near Fitz Park a perfect place to stop for an ice-cream and play before setting off back again - or alternatively sending mummy or daddy back to get the car!