Giddy, with dd1 I would drive. \link{http://www.zsl.org/london-zoo/visit/how-to-find-us,125,AR.html\London Zoo how to find us} and \link{http://www.zsl.org/london-zoo/visit/access-for-disabled-visitors,126,AR.html\Access for disabled visitors}. It's outside the congestion zone, but make sure to plan a route there that doesn't involve going into it (although Londoners - there are no congestion charges at weekends anyway? And if so, a disabled badge holder wouldn't have to pay, but would be more hassle to stop and confirm the badge exists?)
Literally nearest tube station is Camden Town, but all the stations below the zoo, Regent's Park etc are on the Bakerloo line from Paddington. Aylesbury trains come into to Marylebone (also on the Bakerloo line), MK trains come into Euston, which is only a couple of stops away, and Oxford trains come into Paddington.