If my Wailord uses "Giant Wave" this turn, can my opponent's Mew EX's "Versatile" Ability use that attack on the turn following mine?
Yes. The effect of "this Pokemon can't use Giant Wave next turn" applies only to the Pokemon that attacked with it.
Source: Dragons Exalted FAQ (2012-08-23), TPCi Rules Team (2012-08-23)

Relevant cards

Note: All applicable cards may not be listed below.
EX12_88