This Englishman, Irishman and Scotchman song was sung this side of the pond in the 1860s by Sam Collins (died in 1865) and also by George Ware. Quite a few of the later broadside printers gave it but I haven't seen anything earlier than the 1860s. Such and Fortey both in London, Pearson in Manchester, Sanderson in Edinburgh and the Poet's Box in Glasgow and in Dundee. I don't know who wrote it and it's not in Kilgarriff. Pastor was well-known for ripping off British songs but that doesn't mean he did it with this one necessarily. By the 1840s songs were flying back and forth across the Atlantic within weeks of first appearing.

The metre of Gooseberry Tree doesn't seem to match that of any of the 4 tunes in the other.

As it says above, the last bit goes to the chorus of 'In the shade of the old apple tree' so I presume the other verses use the tune of the verse part of that song also. This tune is 1905 so a reasonable presumption is 'Gooseberry Tree' is a much later rewrite of 'Englishman, Irishman'.