Royal Oak is in fact close to the 29th of May as published in Playford. The first eight bars are Playford's first four bars repeated with a some difference in bars four and eight (to accommodate the words). The remainder of the tune is reasonably close to the Playford version but does differ. I notice that in the Cyberhymnal, "Royal Oak" is credited as "17th Century English Tune, arranged Martin Shaw".

You can check the actual Playford version from the link given by treewind. It's no. 155 in their set of tunes.

The version of the 29th of May that Joe Offer linked to differs somewhat from the version in Playford but is recognisably a version of the same tune.

William Monk's tune (which is the one I remember from my childhood) could also have been based on 29th of May, especially if we bear in mind that these tunes would naturally have varied over time.