The National Probate Calendar from 1858 provides a short summary of the will and any letters of administration (admons) which will enable you to purchase a full copy. Birmingham District Probate Registry currently holds the National Probate Calendar on microfiche from 1974 to 1996. They also have a searchable database (“will finder”) covering 1921 to the present. The database is available on a single computer terminal and can be accessed during opening hours (there is no booking available for this.)
The Library of Birmingham will have copies of the Prerogative Court of Canterbury calendars from 1853 to 1857 and all of the National Probate Calendars for England and Wales from 1858 to 1998. We also hold a set of full transcribed copies (not just an index) of wills proven in the years 1858 to 1941 at the Birmingham District Probate Registry, but this does not include Letters of Administration.