Sharp said it was, The BBC project in the 50s was said to be the mopping up of the remnants, Tom Munnelly, working in Ireland from the 60s to the 90s (a full time collector, probably the most prolific in these islands with 22,000 songs to his credit) described his work as "a race with the undertaker". All managed to unearth new material, but each were right in their way. The songs they were collecting were being remembered rather than part of a living tradition. The possible exceptions were the Irish and Scots Travelling communities among which old songs (particularly ballads) were still circulating and being adapted, and new ones were being made and absorbed.
As far as the Irish were concerned, this stopped somewhere between August 1973 and Easter 1975, when it became possible to buy portable televisions at an affordable price. Can't speak for the Scots, but I would guess the same would apply more or less.
Since then, the only way to argue that traditional songs are still being made, absorbed and adapted to serve the communities has been to attempt to either redifine the terms folk, tradition and community, or abandon any definition altogether and claim that everything not sung by a horse is folk and traditional - you choose.
This is not to say that songs yet undiscovered will not turn up in one form or another.
Jim Carroll