I've seen this question be a very hot topic on other boards! However, here's my tuppenceworth. Early in the evening it's quite family friendly; keep in mind that it *is* still Disney but also alcohol is freely available. When I've been there quite late at night it became much more adult, but even then, nothing untoward was happening
The Comedy Warehouse (comedy improvisation and often very, very funny) often makes reference to the fact that children may be present and asks adults to keep this in mind when they shout out their responses and in the Adventurers' Club (a quite whacky and entertaining old colonial club) quite a lot of the humour is only mildly adult by today's standards. Any children present are likely to be 'ribbed' along the lines of having parents who take them to bars; that sort of thing. There's a Jazz club and other music/dance clubs too. A lot will depend on the ages and nature of your children I would guess.
There is usually a good live band on the West End stage and this can be experienced for free. Why not take a trip there, see the band and decide if you like the atmosphere of the place and if not, it's a pleasant stroll to the Market Place.