Just thought it might help to list the ones in Pennsylvania, from census info at familysearch. These are the ones I have found:
- George Aucott c1819/1821 from 1850 census onward, with wife Ann c1817 England then wife Charlotte c1811 England, son Thomas c1843 born England, other children born Pennsylvania
- Saml Aucott/Samuel Aucutt c1815/1816 England, died 1889 Phila PA (married but can't see with wife in census)
- Joseph Aucott c1823 with wife Sarah c1832 born England, children all born in Pennsylvania
- Thomas Aucut c1812 England with wife Mary c1812 England, eldest children born England
c1836, 1838
- John Ocott c1817 England, died 1892 Phila PA?
Children of James and Ann shown at FreeREG were:
William 1813
Cornelia 1817
George 1821
Joseph 1823
FS also has:
Edward 1803
Is the c1804 birthdate for James taken from records in the US that match that age and wife's name (Ann Billings)?
Samuel c1816 with wife Lucy and George c1821 are together in St Margaret with Bishops Fee in 1841.
Thos c1811 and wife Mary are also there.
I don't see Joseph (or John), even using wildcards.
And I don't see your mystery James Aucott in the US or in England in a census!
There's a James Aucutt c1806 in St Geo Han Sq Middlesex in 1841, not born in county, but his wife is Mary, with son George c1831. There's a George with wife Sarah who had children Charles 1861, Frank c1865, Howard A 1866, Mida Hawkins 1868, in Phila PA ... per US censuses, born c1834-6 England. He seems to have fought in the US civil war. Can't tell whose kid he is, though.
A James Aucott was baptised 1800 Market Bosworth, son of Edward and Elizabeth. (There are numerous slightly later baptisms in Stoke Golding to Edward and Hannah, married 1810 - witness Jas Aucott.)
Your James Sr appears to have been the son of Thomas and Mary, who had numerous children but no Edward shows at FS. James Sr and Edward, both of whom baptised children in Stoke Golding, are obviously related, though. For Edward's son to have emigrated with James's sons wouldn't be unimaginable.
I note a young James Aucott in the 1861 and later censuses in Leicester ... born c1846 in Canada(?) or "America" ... also William Aucott born c1848 "America". In the 1850 US census they are with parents Thomas and Mary, in Germantown, who also returned to Leicestershire.
I have been no help at all, just doodling with a tricky name and set of facts, and adding a few details that might help someone else looking at your question!