Activity World on Padholme Road East (near the Showcase Cinema) is probably my favourite.
The main play area is all in one big room so it's easier to supervise children from different age groups. There's a baby/toddler section right next to the main play area.
There's also Big Sky but personally I find it far less friendly than AW. During busy times they also have a policy of only allowing people to stay for a certain amount of time and then they have to leave.
Beware. Activity World are charging £1.50 for a jug of TAP WATER and do not permit own drinks (including water) to be taken in. Total rip off and should be illegal in an environment specifically for young children.