It may not be the first time he told the story, but Shipton does recount it in his autobiography 'That Untravelled World' p85: "As we had done in Africa, we continued to address each other as 'Tilman' and 'Shipton'; and when, after another seven months continuously together, I suggested it was time he called me 'Eric' he became acutely embarrassed, hung his head and muttered, 'It sounds so damned silly'".
It's a good story, but Jim Perrin suggests that it may have been one of Shipton's frequent jokes, just to illustrate how reticent Tilman was - on the basis that Tilman was unfailingly polite and wouldn't have responded in that way. I guess we may never know.