The full OED labels it "colloq. (orig.N. Amer.)", and gives the first British citation in 1966. I think you'd need to have been living in a very dark British cupboard not to know what it means and/or use it as the standard term for that particular activity.
I hope Ewie's right because, at least in AmE, skinny dipping has a slightly different connotation than nude swimming. Both involve swimming naked, but nude swimming implies nude beaches and semi-serious adult stuff, whereas skinny dipping is more of a spontaneous activity. If a group of kids who are fishing in a creek decide to cool off with a swim even though they forgot their swimsuits, I would invariably call that skinny dipping, not nude swimming. If someone goes to a nude beach to swim, I'd call that nude swimming.
Edit (cross-posted with Ewie's post below): Or if I wanted it to sound slightly racy, I'd say nude swimming. Skinny dipping often (though not always) has a connotation of innocence or childishness.