Hi Asheth, we went to this place, Camping de la Mer de Glace. It's slightly out of Chamonix itself, in a village called Les Praz, but there's a lovely little path you can use to walk or cycle into town. We took bikes on the top of the car for this purpose. The views were just stunning, and in the morning and evening you would get these incredibly sunrises and sunsets over the mountains, right from your tent. We loved the hedge divisions so each pitch has privacy.
The only problem - and there always is one - is that it's not far from the local helipad, and if there's an emergency call out a helicopter might need to take off in the middle of the night - that was LOUD. It didn't happen too often though.
Cham is brilliant for young children. Ours were also 11 and 7 when we went and we did loads of activities:
climbing with a trainer several times at the Les Gaillands crags,
the high ropes course (which level you can do depends on height, not age, IIRC),
the summer luge (you have to go up in a ski lift for both of those),
a small amusement park (it's not exactly Alton Towers, but fine for smaller kids),
the train trip to Mer de Glace and the ice grotto there,
the cable car up to 3,800 metres for the Aiguille du Midi (v expensive but worth it),
then we did a few alpine walks too, particularly easyish ones where we could get the cable car up from Les Praz nearby - I know we walked from Flegere to Lac Blanc, and then had ice cream at the refuge there! There's a great map here of all the walks from Flegere. For small children I can recommend walking downwards only from Flegere on the 9 red route through the woods via La Floria restaurant.
We also went in mid-August and caught the Fete des Guides, when the Compagnie des Guides puts on a spectacular display at Les Gaillands for tourists, I think it was free, but donations welcome for old and disabled guides. They were skiing down the rock face, ffs!
Avoid the restaurants in the town centre, particularly sitting outside on the big central square. As we found out to our literal cost, they're rip-off joints.
Hmm, as to the temperature, I can't compare it to the UK because we don't live in the UK and have never camped there. I can well imagine it's colder in Cham at night, but a much drier, brisker cold. We have fairly normal sleeping bags, but we do have camping beds a foot off the ground to keep the rising cold at bay, and do dress very warmly to sleep. But we live in central Europe where -10 in winter is normal, so maybe we're more used to such temperatures.
I really loved that holiday! The only reason we never went back was because it's a tough two-day drive for us each way. But I think it would be a lot more fulfilling for kids than Annecy.
I can't compare it