Dissenters/non-conformists were the way that those who were not members of the Church of England were referred to. This covers a wide range of different groups - in Cornwall, I have a lot of non-conformist relatives who were Wesleyan Methodist, very popular in the area. There were a lot of Independent chapels not part of any particular group. The history of the church says that those who occupied the church during the time of Thomas Sharp were Calvinists:
http://en.wikipedia.org/wiki/CalvinismOn IGI, there's a christening at Crown Street Soho for Eliza Akehurst Sharp, daughter of Thomas Sharp and Eleanor Burt, and a variety of earlier baptisms for them.
Timeline seems to be:
1818: first son on IGI, John Burt Sharp, christened at Spa Fields Lady Huntingdons
(
http://archiver.rootsweb.ancestry.com/th/read/LONDON/2000-08/0967392711 - info on Lady Huntingdon)
1820: first child to be christened at Union Chapel in Woolwich
1832: last child to be christened at Union Chapel in Woolwich
1833: moved to Crown Street Chapel
1834: Eliza christened at Crown Street.
http://www.rootschat.com/forum/index.php/topic,311319.0.html - someone here has already found the marriage license from 1816. They married in Whitechapel - you should try and find the original marriage and see who the witnesses are. There's also a suggestion that he came from or around Great Yarmouth.
It's also a good idea to look at other possible family members attending the same church. Many of the independent chapels had quite small congregations.
For example, also baptised at Spa Fields Lady Huntingdons were:
Children of Francis Sharp and Jane (1788 and 1791)
Children of James Sharp and Elizabeth (early 1800s)
one child to a John Sharp and Charlotte (1814)
Also christened at Union Chapel in Woolwich - no other Sharps, but some children to David Burt and Am(e)y
David Burt and Amy (nee Ward?) may have married in Ipswich in 1819 - the timing is about right.
Another hint:
Evangelical Magazine and Missionary Chronicle lists an obituary in which it mentions someone who went (perhaps around 1813-ish) to Union Chapel, Woolwich, to hear the 'late Rev Thomas Sharp, MA, from Cheshunt College' preach.
Cheshunt College probably refers to Cheshunt College in Hertfordshire, founded by Countess of Huntingdon. It was a non-conformist theological college. You should check with Hertfordshire archives to see what sort of records might have been kept about students.
Google Books has a book written by Thomas, also! He's listed as 'T. Sharp'.
The Heavenly Sisters; Or, Biographical Sketches of the Lives of Thirty Eminently Pious Females: Partly Extracted from the Works of Gibbons, Germont, and Others, and Partly Original : Designed for the Use of Females in General, and Particularly Recommended for the Use of Ladies' Schools