Richard Brinsley Sheridan's The Forty Thieves: A Grand Melo-Dramatic Romance was staged at around the time (or a bit earlier; I haven't found a firm date so far) that Morgiana tunes began to appear. The show certainly involved songs and incidental music (examples can be seen at Levy), though I don't think any are the Morgiana that John Clare knew. There may be a connection; there may not. It seems like a possibility worth exploring further, at any rate.

The Morgiana in... tunes would be commercial spin-offs from Morgiana, I suspect, and adopted into tradition in England and Ireland pretty much simultaneously from the stage, concert-hall or assembly-rooms.