This Guiseppe Tuzi as he was born which became Tuzio then eventually Tuzzio. He came to Birmingham in 1895, was born in 1872 and died in 1943 at 47 Hertford St,Balsall Heath.
Your great grandfather Vengenzino ( Vincent though I think he was known as Sonny) was born in 1902, they messed up his surname :
Births Sep 1902 (>99%)
Zuzio Vengezino Birmingham 6d 61 and he died in 1943 same year as he father.
He was baptised at St Michaels Catholic Church.
Baptism name of Joannes(John) added, Godparents, Lillian Farrell and Guiseppe Tourni?
His mother was Isabella Kate Farrell and she was born 1880 and died in 1956. They were married in 1899 at St Michaels and had 16 living children though about 3 died as children.
That's a nice photo of your great grandfather.
There is no connection to the Liverpool Tuzzio that I know.