I'm no expert on the subject, but about 20 years ago I did pick up a duplicated booklet I couldn't resist entitled "Birmingham Jewry, 1749 - 1914", by Aubrey Newman, then (1980) Reader in History at the University of Leicester and co-founder of the Birmingham Jewish Research Group. It deals mainly with the 1851 and 1871 censuses, which are very well researched. Page 10 notes that the first record of Jews in Birmingham was an unnamed Jewish person met by Daniel West on 29 May, 1749. According to Aris's Birmingham Gazette, a Hungarian, Mayer Oppenheim had set up a glass house in Snow Hill by 1762, and at least eight Jewish Traders were listed in Sketchley's Directory for 1767.
In 1780 William Hutton referred to the Jewish settlement in the Froggery (later occupied by New St Station), branching off Peck Lane. A small house in this street was rebuilt or converted as a basic synagogue, and the garden was used for burials.
In December 1783, Aris's Gazette advertised a plot of land in Granville St, next to "the Jew's Burial Ground", and another was consecrated in 1823 between Bath Row and Islington Row.
The first map below shows both old and new cemeteries as they were in 1839. The old one, off what is now Granville St, was taken over by the railway by 1876, while the new one, south of Islington Row, but also on the western side of the canal, ended up cut back with railways on both both sides, as shown on the second map from 1914. I can't say what's there now.
I think it's a fascinating story, worth more study.
Peter