Which OneWorld airline do you think flies MXP-FCO? I think you'll have to do that on a separate ticket.

As a OW Emerald you can enter the F lounge with one guest when flying OW. You can take 2 guest into CX lounges (but you'll be in F then anyway).

So essentially at JFK and LHR you are dependent on the goodwill of the lounge dragon. And with BA lounge dragons goodwill is in notoriously short supply.