"High, Ho, Diddly Doh" is new to me but, playing the tune in Oz in the early 70s, musos would all sing
Diddly ah, di diddly ah, di ah diddly ah di da twice through for the A part, often without also playing the tune at the same time; only instruments would be used for the B part. This would be repeated whenever the tune was played.

Other examples of mouth music I observed in tunes in Oz were to the Rakes of Mallow and to the Battering Ram.

Rakes was usually known as "Balls up" because of the opening notes of the A part (and I gather this is widely known and not limited to Oz) but, due to events preceding the 1975 National in Sydney, other amendments were accrued. At the end of the descending notes in the phrases of the B part there are three rising notes; in Canberra and Sydney these were routinely replaced by the words "Warren Fahey!" and the tune became widely known (in Oz) as "Planxty Fahey". Warren is one of Oz's great collectors and disseminators of traditional and folk music in Oz as well as a longtime performer and this was a bit of 'piss-taking'.

In the early 70s a house in McIlwraith St, North Carlton (Melbourne) was where a lot of folkies lived and the sessions were legendary and crowded. The opening bars of the A part of the Battering Ram became replaced (in Melbourne) by

Don't piss in the garden in McIlwraith St,
don't piss in the garden in McIlwraith St.
You can piss in the kitchen and piss in the hall
but don't piss in the garden in McIlwraith St.

We're all much older now.

Cheers, Rowan