I don't know about Lloyd, but what Swarb plays with M Carthy here
http://www.youtube.com/watch?v=M3dJHy7mDck
is the first strain of Dorrington Lads, more or less the Northumbrian Minstrelsy version. The vocal tune is simplified.
The only 'Drunken Piper' tune I know is a totally different 2/4 Scottish march - there may be other 'Drunken Pipers' but they're not Dorrington Lads.

As a very strange coincidence Martin Carthy later recorded the two tunes from the Dixon book I mentioned above on:
Broken Ground
Waterson:Carthy
Topic TSCD509
[Adam A Bell; Dorrington Lads]

No idea whether he made the connection to Byker Hill, I think that medley was Eliza's idea.