Hello Suzie....
Sorry I see others are on the trail while I have been typing.... will post anyway
There isn't an Arthur Bezant on the Nottinghamshire FHS indexes and as you say apparently not on BMD indexes.
This is my theory, bear with me as it gets complicated and you will need to follow the census etc.
As on the 1871 Arthur gives his occupation as Gunmaker - I think the key is this link.....
http://www.doublegunshop.com/forums/ubbthreads.php?ubb=showflat&Number=145842&page=1.
1841 = Piece 1015 Book/Folio 4 / Page 1 = Address Thoroughfare, Halesworth ,Suffolk
Simon Bezant - Gunmaker
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1851 = HO107 piece 2131 folio 189 page 39 - 2, Parliament Place, St Mary, Nottingham
BEZANT, Frederick S - Head - 30 - Gun Maker - born Ringsfield, Suffolk
BEZANT, Mary Wife - 40- Gun Maker's Wife born Onehouse, Suffolk
BEZANT, Alfred Lodger Unmarried - 19 - Joiner Ringsfield, Suffolk
ROUT, William H - Step Son- Unmarried aged 15 - bornPainter - Stowmarket, Suffolk
READ, Edwin Lodger Widower - 55 - Potter- born Burslem, Staffordshire
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1851 = Piece 2130 /Folio 309 /Page 27 - Manvers Street, Snenton, Nottinghamshire
Thomas Doughty - married aged 68 - Frame Work Knitter - born Sutton Bonnington, Nottinghamshire
Elizabeth Doughty - wife aged 64 - born Leicestershire.
Eliza - dau - aged 34 - born Loughbro, Leicestershire
Ann dau aged 32 - born Loughbro, Leicestershire
SARAH TUCKWOOD, daughter, Widow, aged 30 - born Loughbro, Leicestershire
ARTHUR TUCKWOOD, grandson - aged 7 - Scholar - born Nottingham
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(I think he is Frederick Simon Bezant)
1861 = Piece 2184 Folio 54 Page 23 - Potters Hill, Aston, Warwickshire
BEZANT, Frederick S - Head - 41 - Gun Maker And Draper Assistant Employing 1 Man - born Ringsfield, Suffolk
BEZANT, Sarah Wife Married- aged 40 - born Loughton, Leicestershire (Loughboro' ?)
TUCKWOOD, Arthur - Apprentice - Unmarried - aged 16 (born abt 1845) Gun Maker - born Nottingham
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There is a Marriage SeptQ 1843 for John Tuckwood and on the same page is Sarah Doughty - but this marriage isn't on the Notts FHS marriage index, nor is there a baptism for Arthur Tuckwood.
Births Sep 1843
Tuckwood Arthur - Nottingham 15 583
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If Sarah Tuckwood did marry Frederick Simon Bezant I can't see a marriage for them?
I hope it gives you something to go on.
Sheila.