Both deal with encounters at sea, but they aren't really related. 'The Royal Oak' (Roud 951: also 'The Twenty-Fourth of February', 'Captain Mansfield's Fight With the Turkes at Sea' etc) seems likely, as A L Lloyd noted in The Penguin Book of English Folk Songs, to have been based on an incident of 1669.

'The Bold Princess Royal' (Roud 528) concerns an incident of 1798 (for details, see Marrow Bones). It was a post from a one-off visitor here, in a thread otherwise devoted to the (unrelated) 'Princess Royal' dance tune frequently attributed to O'Carolan, that put me on the track of that. Actually the real-life event was more dramatic than the song based on it.

The Marrow Bones text is in the DT: THE BOLD PRINCESS ROYAL (2). Like most versions found in oral tradition, it's quite close to broadside editions (for some of which, see Bodleian Library Broadside Ballads: The Bold Princess Royal).

Peter Bellamy's recording of 'The Fourteenth of February' originally appeared on his LP Mainly Norfolk. I haven't heard it in years, but I'm fairly sure that it was an arrangement of either Harry Cox's or Walter Pardon's versions. Both were also fairly close to the broadside form.