Newlands Corner has big car park, kids play area, great view, cafe and is a short drive to Shere if you wanted a pretty village/pub lunch. The riding along the NDW isn’t that inspiring as you are in the trees and it’s out and back as any loop involves a descent and climb so not really family friendly. Newlands is very busy and quite a bit of dog mess sadly. If you skip the cycling idea I can send you a nice family friendly 1hr walking loop from Shere with pub stop options.
If you went to Wisley you could ride the canal / river towpath, you could do this from Guildofrd too (towards Godalming)
Another idea would be Frensham Ponds, quite nice with the lake and not so hilly but it’s quite sandy so not ideal riding ground.
I think TMH’s suggestion is probably the best one so far