In history, a cordwainer made new shoes from new leather, compared with a 'cobbler' restricted to repairing shoes. One famous family of cordwainers is the Chamberlain family (see attached newspaper extract).
I've gathered together the list below from newspaper notices (mostly Aris's Gazette, Birmingham Journal and Perrys Bankruptcy Journal) The majority of details listed are from announcements of business failures, insolvency and bankruptcy. Some are from announcements of an absconded apprentice. Where no address or further details are given, these were shown in newspapers simply as 'Birmingham' 'cordwainer'. A few are located in the suburbs of Birmingham.
1700s
21/6/1756 William Bratt, Sutton Coldfield
4/2/1760 Thomas Swingler
25/8/1766 Mr Cattell, fronting Steelhouse Lane and Slaney Street, on one side of Galton's warehouse
25/5/1772 Joseph Heath
24/4/1786 Edward Hand, Hill Hook, Sutton Coldfield (had apprentice Thomas Stephens)
3/3/1788 Joseph Caddick (had apprentice Joseph Bunn)
23/6/1788 George Bullin, High Street, large shop to front of street
15/2/1790 Thomas Baskerville
8/1/1791 Thomas Jones, Moor Street
19/3/1792 Francis Green, Great Charles Street
17/10/1791 Thomas Collins
27/8/1792 William Morris (had apprentice Thomas Heathcock)
5/11/1792 Robert Dedwick
5/11/1792 Thomas Collins
4/5/1795 William Benham
4/9/1795 John Ingall, Colmore Street
28/1/1799 Isaac Ashford, Hangers Hill, Birmingham
15/2/1799 Mr Shaddock, fronting Dale End
1800s
26/1/1801 Robinson
21/5/1804 John Colwell
27/3/1809 Thomas Martin
13/3/1809 Thomas Cartwright, No. 1 Belmont Row (had apprentice James Sturges)
26/7/1813 Joseph Shenton, Union Street
16/8/1813 Joseph Pottinger
17/9/1810 Joseph Penny, Lichfield Street
19/10/1818 Joseph Penny, Lichfield Street
19/10/1818 Thomas Jones, Bull Ring
18/3/1822 Edward Field, Sutton Coldfield
7/10/1822 Joseph Lewis, Little Hampton Street
17/2/1823 William Madison, Deritend
26/3/1831 James Lowe, Moor Street
28/10/1836 Edward Corns, Pershore Street, then Great Barr Street, then Ryland Street, late Bordesley Street
12/11/1836 William Brindley, formerly of Pope Street, then of Cardigan Street Aston, then of George Street, late of Barr Street Hockley
27/3/1837 William Edwards
27/3/1837 Charles Kenyon, Aston
29/5/1837 Joseph Halward
19/2/1838 James Gough, Alcester Street, Aston
7/4/1838 John Jones
5/11/1838 Thomas Greatrex, Coleman Street
12/11/1838 Nathaniel Brewer, formerly of Deritend, late of Holland Street
15/6/1839 Benjamin Higgs
21/11/1840 John Hawkes
16/1/1841 Frances Robinson
9/8/1841 Charles Harrison
15/11/1841 Charles Robinson, formerly of 173 High Street, Deritend, then Warwick Street, lately Allison Street
12/2/1842 Charles Robinson
4/4/1842 John Smallman, formerly of Yardley, late of Sparkbrook
14/11/1842 Samuel Baker
14/11/1842 William Jones
20/7/1844 Joseph Oliver
1/2/1845 Luke Wickstone
10/11/1845 Benjamin Burton, Handsworth
10/11/1845 George Simcox, Handsworth
27/6/1846 Joseph Carding, Cherry Street
13/11/1848 William Powell, Dudley Street, lodging at Sherbourne Road, Balsall Heath
26/11/1849 Joseph Green, Harborne
15/10/1850 Joseph Brooks, journeyman cordwainer, lately 9f 7 Digbeth (lodging), once in Great Colmore Street and Latimer Street
28/2/1851 William White
9/8/1851 James Roberts
15/6/1853 William Pinkiss, Summer Lane
15/6/1855 John Haines
4/5/1861 John Kimberley, Kings Norton
14/9/1861 John Evan, Manchester Street
1/4/1865 T Winner, Kinner, Birmingham
2/9/1865 James Edmonds
16/6/1866 William Hulston, Stafford Street
20/4/1867 Elijah Blackburn, Icknield Street East
7/11/1867 John Hyde, Bell Barn Road
28/5/1869 James Chilton, Reeston Street, Birmingham
20/4/1878 George Bedford
Cordwainers at work (not Birmingham)

Source : British Newspaper Archive