Your doctor is right. You take it at night to fall asleep. Please don't take it on the plane as you wont get enough hours in to justify it and some people report more vivid dreams with it. You don't want to be the "inflight entertainment" lol!
Definitely don't take it during the day!
Just be very wary of my nap advice. I think they're old enough to be woken up and understand why they need to get going. When I worked, I noticed that my body "registered" a full nights' sleep at 4 hours (that doesn't mean I was fully rested). They may be so wound up by the time they got to the hotel, they wont want to anyway.
Tell the hotel that you want to check in early and if it's possible. Call them once you arrive and remind them.
London has so much on offer that you can figure out the right balance between some activity that would need "awake" concentration but lively enough to keep their brains active. Just being around a lot of people works for me. Once I took my kids to the pool.
All of you keep drinking. Flying dehydrates and those symptoms can make jet lag worse. You probably don't want to give them caffeine but if they're really dropping off, green tea is a good compromise.