I've been going back over the Green family, and its all starting to get very complicated!
Joseph was convicted 20 Mar 1832, so this may or may not be a coincidence:
11 Feb 1832, St Margaret
Joseph Boulton bachelor & Ann Green spinster by banns. Wit: William Jordan.
Joseph signed, Ann made her mark
08 Feb 1833, St Margaret, Emma d/o Joseph & Ann Bolton, Mill Street, FWK
18 Jul 1834, St Margaret, Thomas s/o Joseph & Ann Bolton, Woodboy Street, FWK
The letter you mentioned in your opening post is from the Home office Petitions series, but I think some of the file is missing as the one for my Leicester convict is about 12 pages, so if this is your Joseph there may also have been a letter from Ann. Also I know it would be impossible for Thomas to be Joseph's son, but an Ann Bolton, widow, married Thomas Marvin, widower at St Margaret's in 1837, so it's certainly worth looking into further.
We also had Ann Boulton/Bolton marrying Thomas Green in 1842, and these seem to be their children:
GREEN, EDWARD BOULTON 1844 M Quarter in LEICESTER UNION Volume 15 Page 93
GREEN, ELIZABETH BOLTON 1846 J Quarter in LEICESTER UNION Volume 15 Page 109
GREEN, HARRIET BOLTON 1849 M Quarter in LEICESTER Volume 15 Page 90
GREEN, MICHAEL BOULTON 1854 M Quarter in LEICESTER Volume 07A Page 160
GREEN, THOMAS BOULTON 1856 M Quarter in LEICESTER Volume 07A Page 170
GREEN, JOSEPH BOULTON 1858 S Quarter in LEICESTER Volume 07A Page 145
GREEN, WILLIE BOLTON 1862 S Quarter in LEICESTER Volume 07A Page 171
In 1841 Ann is living with Eliza and 1-year old George Bolton/Boulton, and a George Green s/o Thomas & Ann was baptised at the same time as Edward in 1844.
Then to complete the hat-trick there is this:
9 Sept 1844, St Margaret
George Green of full age, bachelor of North Gate Street, Father: Samuel Green FWK
Elizabeth Boulton of full age, spinster of North Gate Street, Father: William Boulton FWK
Wit: Frederick Turney (?) & Charlotte Boulton
Their children seem to be
GREEN, ELIZABETH BOULTON 1846 D Quarter in LEICESTER UNION Volume 15 Page
GREEN, ANN JANE BOULTON 1849 J Quarter in LEICESTER Volume 15 Page 137
GREEN, EMMA BOULTON 1861 J Quarter in LEICESTER Volume 07A Page 182
Plus there are three I haven't matched with parents yet:
GREEN, SARAH ANN BOULTON 1845 M Quarter in LEICESTER UNION Volume 15 Page 123
GREEN, SARAH ANN BOULTON 1851 S Quarter in LEICESTER Volume 15 Page 96
GREEN, WILLIAM BOULTON 1853 J Quarter in LEICESTER Volume 07A Page 202
I have no idea how many different Green or Boulton families are involved here, but good luck with trying to figure it out !!