There are a lot of Underwoods mentioned in directories in birminghamin the period, but none fit the names you have given (other possibly than ann). In case they are of interest I list them (1815-1855) below
1815-18 Underwood Joseph, gilder in general, Mount-street
1818 Underwood William, gilt toy maker, Navigation-street
1823 Underwood Joseph, victualler, 44, Bull-street (Rainbow & Beehive pub)
1818-33 Underwood John, gilder, 43, Mount-st.
1839
Underwood Joseph, haberdasher, 63, Fazeley street
Underwood Joseph, gilder, Charlotte street
Underwood Thomas, lithographic printer, 88, New street
Underwood Francis, cabinet maker, 35, Sun street
1841
Underwood Francis, cabinet maker, 89 Bristol st
Underwood Thomas, lithographic printer; 6 Temple st
1845
Underwood..Miss Ann, ladies' school. 24 Sandpits
Underwood Mrs. Ellen, guard chain maker. Charlotte st
Underwood Thomas, lithographic draftsman & printer, 78½ High st
1849
Underwood .Ann, school. Sandpits
Underwood Francis,cabinet maker,35,Sun .st
Underwood Henry manufr. of russia, morocco and all kinds of leather pocket
books, writing, dressing, travelling, shaving, card, cigar, jewel, medal, spectacle,
and :surgeons instrument cases, 29½ Smallbrook St. ; h. 163 Bristol street
Underwood Joseph, pocket book & writing case maker, l65 Bristol street
Underwood Joseph, assistant overseer, 126,Vyse street
Underwood Mary Ann, haberdasher, 120,Hockley hill
Underwood Francis, cabinet maker, 36 Sun street
Underwood Thomas, lithographic draughtsman, engraver,& printer, 78½ High street
Underwood William, grocer, Moseley st.
1855
Underwood Ann, ladies school, 47, LowerCamden st
Underwood Caroline, modeller of wax flowers,13, Union passage
Underwood Charles, coal dealer, 22, Park st
Underwood David, haberdasher,89 Edward street parade
Underwood Eliza (Mrs.), Falcon hotel, 101 Newhall street
IJnderwood Eliza (Mrs.), milliner, 7 High street, Deritend
Underwood Francis, cabinet ma. & upholsterer, 16 Sun st
Underwood George, agent to A. Fullarton & Co., 3, Bennett's hill; h. 9, Highgate
Underwood Henry, baker, 34 Springfield street West, Lower King Edward's road
Underwood John, shopkeeper 21,Springfield st west
Underwood Joseph, cabinet maker,11 Holloway head
Underwood Joseph, manufacturer of flexible purses, pocket books & patent expanding purses, 142 Irving st. Bristol st
Underwood Joseph, victualer., Falcon,Family & Commercial Hotel, 101, Newhall st
Underwood Thornas, lithographic draughtsman, engraver,stationer & print publisher, l3 Union passage, & wholesale warehouse, Castle street, High St
Underwood William., tobacconist
mike