Here's the only reference I could find concerning the styling - particularly the headlights:
"Longtime sportsman Toly Arutunoff owned a Cunningham with a body reminiscent of a C2R but with quad headlites rimmed by hardware from an early Corvair or Lark. It was most likely a C3 road car that had missed the boat to Vignale for their rather dated and dowdy coachwork, or a rebody on a scrapper. I had photo reference of it (now unavailable) and believe it also had the soft top of an early Aston Martin. As Toly was a collector of orphans and not too concerned about "flipping" them for a quick buck, I trusted that he was not inventing stories to cover its (lack of) provenance."
That doesn't give much of a frame of reference for when the car was (re)bodied, and in any case you didn't originally ask that. I'm not sure I agree with the writer's assessment of the headlight hardware coming from a Corvair or Lark, but that would put the rebody in the early 1960s.