I would be looking at both sides of the river for the burial. For Liverpool Ancestry has great coverage of the local authority and RC cemeteries.
For Wirral
Bebington Cemetery, register of burials in the consecrated ground, Roman Catholic ground, and unconsecrated (Nonconformist) ground, 1868-2006 [1868 to 1960]
https://familysearch.org/search/catalog/1026599?availability=Family%20History%20LibraryBirkenhead Cemetery (Flaybrick), register of burials and graves in the consecrated (Church of England), unconsecrated (Nonconformist) and Roman Catholic grounds, 1863-1990 [1863 to 1959]
https://familysearch.org/search/catalog/1027117?availability=Family%20History%20Library“Frankby Cemetery, register of burials in consecrated and unconsecrated division, 1940-2004” [1940 to 1961]
https://familysearch.org/search/catalog/1027065?availability=Family%20History%20LibraryGrange Cemetery/Hoylake & West Kirby Cemetery
https://familysearch.org/ark:/61903/3:1:3QSQ-G92H-FYCT?i=175&cat=1027047C of E Landican burials 1934-1950
https://familysearch.org/ark:/61903/3:1:3QS7-L924-PHFY?i=673&cat=1027065C of E Landican burials 1951-1962
https://familysearch.org/ark:/61903/3:1:3QSQ-G92H-N35D?i=5&cat=1027117RC Landican burials 1935-1962
https://familysearch.org/ark:/61903/3:1:3QSQ-G92H-N34R?i=167&cat=1027117Non-conformist Landican burials 1934-1962
https://familysearch.org/ark:/61903/3:1:3QSQ-G92H-NQPR?i=332&cat=1027117“Wallasey Cemetery, register of burials in Roman Catholic ground, unconsecrated (general) ground and consecrated (Church of England) ground,, 1883-2006” [1883 to 1973]
https://familysearch.org/search/catalog/1027047?availability=Family%20History%20LibraryFor Bootle
England, Lancashire, Bootle, Bootle Cemetery, register of burials and graves, 1912-2000 [Records may cover two pages, left page and right page]
https://www.familysearch.org/search/catalog/1910606?availability=Family%20History%20LibraryFor Southport
“England, Lancashire, Southport, cremation index, 1959-1998” [includes more than this description]
https://familysearch.org/search/catalog/1910612?availability=Family%20History%20Library“England, Lancashire, Southport, Duke Street, burial and grave register, 1866-2000”
https://familysearch.org/search/catalog/1910607?availability=Family%20History%20LibraryFor Birkdale
England, Lancashire, Birkdale, Liverpool Road, burial and grave registers, 1901-2000
https://familysearch.org/search/catalog/1910610?availability=Family%20History%20LibraryC