I've checked both the FREEBMD and FINDMYPAST sites and there are no births for a Thomas Phelps registered in the Jul-Sep quasrter of 1861.
There are however just 2 in the next quarter, Oct-Dec. (Child might have be born Aug but registered Oct).
Births Oct-Dec 1861 PHELPS Thomas Herbert Newent Vol 6a Page 209 PHELPS Thomas Nowell Taunton Vol 5c Page 386
The first of these is in Gloucestershire, the second in Somerset.
As Sandra has suggested, you really need to track down the marriage record in the States, as that should give more details of parents. We might them be able to identify Thomas and Emma for definite on census records.
Other information on the family such as the names of their children and Thomas's occupation MIGHT also give some clues as to the correct Phelps/Yarrow families here in the UK. Most likely to be helpful if any of these are unusual.