See 'Desert Dancer's' link for a discussion of the practicalities of underwater drilling.

My shelves don't run to naval history at all, but both Nicholas and Pryor can be found in facsimile at Google Books. La Roncière is at the Internet Archive; the page reference given above seems to be wrong. The relevant (and very short) passage appears on page 147:

'Suivant une autre version, ce fut un plongeur qui perfora la carène.'

The reference given is to 'Raoul de Diceto (an English chronicler, I think), col. 662'. It's rather a big jump from 'a diver who holed the hull' to 'a little boy swimming underneath her with an auger and drilling a hole in the bottom': perhaps Rodger had other sources that he didn't specify. It's also possible, of course, that he may inadvertently have conflated the reference with the quite well-known 'Golden Vanity' story, though we might expect better than that from an historian.