1851 census has 128 Bold Street Liverpool William Bridson Ironmonger 33 b. Liverpool HO107/2180
Others are
Sarah Jane wife 24 b IOM
Margaret daughter 2 b. Liverpool
Ann Prowse Bridson daughter 3m. b. Liverpool
Robert Bridson Prowse nephew 13 b. Toxeth Park
As there is a Jan 19 1850 marriage of William Bridson to Sarah Jane Gelling in Andrea.
Margaret may have died and William remarried.

In 1861 RG9/2686 122 Faulkner Street Liverpool
William 43 Master ironmonger
Sarah Jane 34
Ann Prowse Bridson 10
Arthur Bridson 1
John Bridson 7
Margaret Bridson 12
Mary Brown Bridson 4
Sarah Jane 3
Jane McKean 59 servant
Ellen O'Brian 20 servant
All b. Liverpool except Sarah Sr.

In 1871 Sarah Jane is a widow (b. Douglas) aged 42 and is an ironmonger. She is a visitor at No 8 Carlton Terrace St. Meolse Cheshire with Sarah J 12
Lucy 3
Arthur 9
Lister (sic) Liza? 4

I did not find William or Edward Bridson in Tranmere in 1871.
I am using electronic data, so I miss wrongly transcribed data.