The Pudding Brook in Wiltshire starts north of the A4 between Corsham and Chippenham, it flows under the road and the railway line before joining the River Avon south of Chippenham.
Regarding the name, there is definitely one other Pudding Brook, near Birmingham. Thomas Anderton wrote in 1900 of his childhood:
I am not the only person living who remembers "Pudding Brook" and "Vaughton's Hole." The name of "Padding Brook" was, in my boyish days, given to a swampy area of fields now covered by Gooch Street and surrounding thoroughfares. Pudding Brook proper was, however, a little muddy stream that flowed or oozed along the district named and finally emptied itself into the old moat not far from St. Martin's Church.(1)
Here Pudding has become Padding which could link it to the Anglo Saxon paddaneig meaning toad-meadow or frog island. Padde meaning toad or frog. (Paddaníeg [] f (-e/-a) toad-meadow, frog-island also Padde [] f (-an/-an) toad, frog). (2) It has also been suggested that Pudding could be from the Modern English topographical term for sticky muddy soil.
1 - https://billdargue.jimdo.com/placenames-gazetteer-a-to-y/places-v/vaughtons-hole/
2 - From Old English The Portal to the Language of the Anglo Saxons(http://old-engli.sh/index.php)