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Yes try RCI from Southampton the new ship will have more than enough to keep them happy including there own clubs great shows ect.
Absolutely - RCI newest ship Independence is out of Southampton - or try one of her 'sisters' Freedom or Liberty out of Miami.
We regularly take our son aged 6 (nearly 7) on cruises and I agree that Royal Caribbean are fantastic for the children (and the adults). He loved the kids club and participated in lots of fun stuff around and about the ship. The pool and hot tubs are a magnet for similar aged children and they have a whale of a time. The food is excellent with lots of choice. We have sailed Navigator from Southampton - recommended but it is sailing from Europe next year as the Independence is now sailing from UK. Ship looks great. My only advice would be book a balcony so that when the kids go to bed you can sit out with a glass of wine (or 2).
However Independence has been so successful that every 3/4 berth cabin has been sold out for the entire summer season for several months now! Bookings for the 09 summer season will go on sale in April so suggest you get in there quickly.