My kids were doing 20 miles a day at that age. These were day rides, setting out from camp sites and returning. We weren't carrying much weight. I agree with everything that's been said, particularly the crucial element of the youngster's state of mind. Enthusiasm (even bought by the promise of ice cream) is essential.
I've had an extra thought, though. My kids were similar in age, and so had someone to relate to (in performance; at cafe stops; at play times; etc). The OP doesn't say whether this 10 yr old has any mates similar to his/her age, or not. If not, I'd recommend hiring one for the trip.
10 year old - how far?
Re: 10 year old - how far?
It does depend on the child and terrain. We started YHA touring when my lad was nearly 9, and managed up to 45 miles a day across mid Wales. The harder the climb, the faster he went. Which is just annoying 
A bit like finding his panniers full of interesting rocks that he had collected as we went past the mines. We did some training but not too much. Just remember to have loads of energy to hand. It is a great experience.
A bit like finding his panniers full of interesting rocks that he had collected as we went past the mines. We did some training but not too much. Just remember to have loads of energy to hand. It is a great experience.
Re: 10 year old - how far?
I rode the Great Weston Ride at the weekend, a 56 mile basic route or extras taking it up to 70+ miles. After the 4 mile mark, climbing a hill I got overtaken by a child on his Islabike 
I did see him and his father again a few miles on at the bottom of a gentle descent, being 15 stone does have it's advantages I'm assuming they did the 56 miles but looking at his bike he wasn't a newbiem, you don't spend money on an Islabike and hang some Campag off it for a beginner.
I did see him and his father again a few miles on at the bottom of a gentle descent, being 15 stone does have it's advantages I'm assuming they did the 56 miles but looking at his bike he wasn't a newbiem, you don't spend money on an Islabike and hang some Campag off it for a beginner.