Out of 77 references in the Roud Index, most are North American, but there are a sizeable number from Britain. The earliest collected ones seem to date only to the late nineteenth century, but given the wide distribution - from the Isle of Wight and Cornwall all the way up to north-east Scotland (where I assume the Grieg-Duncan text was collected) - it's likely the rhyme is considerably older.

Just as 'Miller Boy' seems to be the most common North American title, 'Jolly Miller' is far and away the most common British title.

'JOLLY MILLER' (England, 1985)
Opie, The Singing Game (1985) pp.314-316

'JOLLY MILLER' (nine versions from around England)
Gomme, Traditional Games of England, Ireland & Scotland 1 (1894) pp.289-293

'JOLLY MILLER' (England)
Gomme & Sharp, Children's Singing Games 2 (1909) pp.6-8

'JOLLY MILLER' (England : Yorkshire : Bradfield, 1907)
R.A.A. Gatty MS collection (Birmingham Ref Lib 661164 11R 20) n/bk 3 p.31 / n/bk 2 p.17

'JOLLY MILLER'
Cecil Sharp MSS, Folk Tunes p. 1705 (England : Derbyshire : Winster, 1908)

'JOLLY MILLER'
Kerr's Guild of Play (1912) p.3 (versions a & b) (Scotland?)

'JOLLY MILLER' (England : Hampshire)
Gillington, Old Isle of Wight Singing Games (1909) p.7

'JOLLY MILLER' (England : Northumberland : Cambo, 1920s)
Bosanquet, In the Troublesome Times (2nd edn. 1989) p.130

'THERE WAS A JOLLY MILLER' (Scotland)
Greig-Duncan MSS (Gm 2.126c)

'JOLLY MILLER' (England : Yorkshire : Sheffield)
Northall, English Folk-Rhymes (1892) p.366

'JOLLY MILLER" (England)
Yeatman & Hall, On the Green (1894) p.17

'JOLLY MILLER" (England)
Hinkson, Victorian Singing Games (1991) p.12, 25

'JOLLY MILLER' (England, I believe)
Kidson, 100 Singing Games (1916) p.115

'THERE WAS A JOLLY MILLER' (England, I believe)
Hornby, The Joyous Book of Singing Games (c1913) p.60

'JOLLY MILLER' (England)
Walter, Old English Singing Games (1926) p.30

'JOLLY MILLER' (England)
Holbrook, Children's Games (1957) pp.100-101
Previous source: Sharp, Children's Singing Games

'THERE WAS A JOLLY MILLER' (England : London)
Douglas, London Street Games (2nd edn., 1931) p.41

'JOLLY MILLER' (England : Cornwall)
Courtney: Folk-Lore Journal 5 (1887) pp.57-58