I know it's a while since you posted your request, but I've just spotted it.
In 1919, a death in Birkenhead would almost certainly have resulted in a burial in Flaybrick Cemetery, which was the first municipal cemetery to open in Wirral. There is a possiblity that there may have been a Church burial. All cemetery records and grave registers are kept at Birkenhead Central Library. If you want to visit yourself, give them a couple of days notice, and the librarian will book a micro-fiche reader for you. If you want the library to undertake the search for you - the first half hour is free, then there is a charge of £10.50 for the next half hour.
The address for the library is:
Birkenhead Central Library
Birkenhead Wirral, CH41 2XB
Tel: 0151 606 2665
Opening hours: Mon & Tues 9a.m. - 8p.m.
Thurs 9a.m. - 8p.m.
Fri 9a.m. - 5p.m.
Sat 9a.m - 1p.m. closed for lunch 2p.m. - 5p.m.
Hope this is useful. P.S. The lady I spoke to was very nice!
Sue Thompson, Liverpool