The only thing I can see for the full name you posted is on family search is below, it also appears on Ancestry.
Plenty of Thomas White's. This suggests he was born in Nottingham and moved to Shropshire.
How do you know he was born in Shropshire and his middle name was William?
Great Britain, War Office Registers, 1772-1935
Draft Registration
Thomas William White
Age: 16
22 Mar 1826
Oldbury, Shropshire, England
Birth Year (Estimated): 1810
Birthplace: Nottingham
Colin