The TfL website doesn't clarify if does or not. However one of the railcards that can be set on oyster (HM Forces) also lets you buy the discounted child travelcards which would imply that it does.
Any adult ticket, Oyster or contactless card is valid for the purposes of under 11s travelling, if that is the question being asked (I'm not sure).
F&F railcards cannot be loaded on an Oyster card, so there is no discount available here. Because it can't be put on an Oyster card you also can't get F&F discounted child-rate tickets when using an Oyster card; both tickets must be magnetics.
However, as opposed to purchasing child-rate travelcards for the 11-15yo I would suggest getting an Oyster card and then asking a member of staff to add the 'Young Visitor' Oyster discount for the duration of your stay; this way you get the flexibility of Oyster PAYG and the discounted travel options. It can be done anywhere in zone 1 at either a ticket office or by any* CSA at a ticket machine.
*That's if the CSA has the relevant login, which we've all been told to arrange.
(Hopefully this reply makes sense; I've tried to address both the OP & Deerfold's questions/comments in one - let me know if not.)