If we are to include offshow holdings, London is definitely not in the running as Slimbridge is heads and shoulders above all competition once that is factored in, even with the collection having deteriorated somewhat in recent times. Focusing solely on onshow holdings, I suspect only Paignton is ahead.
In total, London has 17 amphibian species of which 14 are onshow, but all of the remaining three, as cerperal says, can be seen in the distance as there are windows behind certain tanks into onshow rooms. That 17 is comprised of 3 species of caecilian, 5 species of newt or salamander and 9 species of frog or toad.