Thanks to Chrissie, vectis and Noreen for reviving this topic.
There are several threads discussing the song, and it seems to be a matter of luck as to which one turns up in a Google or other search.

As I live in Bebington in Wirral I can state that there isn't any location known as "Bebington Bush", and to the best of my knowledge there never has been. As a mainly suburban area it wouldn't have been a prime spot for a street entertainer to work. So far as I've been able to discover all the memories of Seth Davy are associated with Liverpool, and although the song does mention New Brighton Pier I haven't yet seen any evidence that Seth Davy performed there.

The "Pivvy" could be a reference to the Pavilion, but some singers sing the words as "Tivvy" - a reference to the Tivoli Palace of Varieties in Lime Street, Liverpool.

I did mention in another thread on this song Lyr Req: Whiskey on a Sunday that the late Fritz Spiegl had found an old magic lantern slide showing a street scene outside the Bevington Bush Hotel from the late 19th century. This shows an elderly black man seated on some sort of box playing a set of jig dolls on a plank and surrounded by children. I've now seen the original slide which is labelled "Davy" and I hope to be able to publish it soon with the owner's consent. I'm sure that Glyn Hughes must have seen this slide, or a picture based on it, when he wrote the song since the scene exactly matches the first verse:-

"He sat at the corner of Bevington Bush
Astride an old packing case
And the dolls on the end of his plank went dancing
As he crooned with a smile on his face."

Matthew Edwards