I'm thinking specifically of "Here We Go 'Round the Mulberry Bush" and "Pop Goes The Weasel." The thing is, mulberries grow on trees, not bushes. Where did this idea come from, and why was it such a popular element of songs for children in the late 19th/early 20th centuries?
Your question is based on a false premise: mulberries (or morus) can grow on bushes as well as trees.
The only differences between a tree and a bush are the size and growth pattern. Bushes are shorter with multiple stems and no obvious trunk. Some varieties of mulberry can easily be trained to become bushes or trees.
All this means that the occurrence of mulberry bushes in European folk song is perfectly correct.
There are at least 2 theories to the origins. According to historian R. S. Duncan, a former governor of England's Wakefield Prison, the song originated with that 420-year-old institution's female prisoners, who were exercised around a mulberry tree. A sprig was taken from Hatfeild Hall (Normanton Golf Club) in Stanley, Wakefield, and grew into a fully mature mulberry tree around which prisoners exercised in the moonlight.
The mulberry tree died during 2017 and was cut down and removed on 7 May 2019. Cuttings were taken during the 1980s and have grown into mature trees. Further cuttings taken from these trees will be replanted at HMP Wakefield to replace the mulberry tree.
Another possible interpretation of the rhyme is that it references Britain's struggles to produce silk, mulberry trees being a key habitat for the cultivation of silkworms. As Bill Bryson explains, Britain in the eighteenth and nineteenth centuries tried to emulate the success of the Chinese in silk production but the industry was held back by periodic harsh winters and mulberry trees proved too sensitive to frost to thrive. The traditional lyrics "Here we go round the mulberry bush / On a cold and frosty morning" may therefore be a joke about the problems faced by the industry.
As for 'Pop goes the weasel' that's about pawning (pop) your Sunday best coat (weasel and stoat) mid week to spent the money on food and booze down 'The Eagle Tavern'.
Sources: R. S. Duncan, Here we go round the mulberry bush' The House of Correction 1595 / HM Prison Wakefield 1995 (Privately published, 1994).
Bryson, B. (2011). At Home: A Short History of Private Life. Black Swan. p. 536.