When Robert Patterson married at All Saints, Newcastle on 22 March 1813, on the records I have seen his wife was noted as Eleanor or Elizabeth Dunn.
If he was the one born below he would be 1 week short of his 15th birthday, unusual but not impossible;
Name Robert Paterson – Parents William, a waterman & Elizabeth
Birth 29/03/1798
Baptism 06/05/1798 All Saints, Newcastle-Upon-Tyne, Northumberland.
The only record I can see for children for Robert & Eleanor All Saints is your William 28 Aug 1814, his father noted as a Labourer.
However noted below there are a number of baptisms for a Robert & Elizabeth, abode New Rd, Robert a waterman. All at Newcastle upon Tyne : All Saints
Elizabeth PATTERSON Baptism 03 Nov 1816 Northumberland
Robert PATTERSON Baptism 20 Aug 1821 Northumberland
Mary PATTERSON Baptism 13 Oct 1823 Northumberland
There may be more on Freereg & Ancestry by coincidence John & Cuthbert Patterson had children at All Saints in the same period both noted as a watermen
Their father William Patterson a waterman. Older brothers of Robert.
Cuthbert PATTERSON Baptism 04 Oct 1789 Northumberland Newcastle upon Tyne : All Saints : Bishop's Transcripts
John PATTERSON Baptism 23 Mar 1788 Northumberland Newcastle upon Tyne : All Saints : Bishop's Transcripts
So maybe by 1816 Robert progressed from Labourer to Waterman and Eleanor used the name Elizabeth.
I’m not sure if any of this helps but do you have any later information on Robert or Eleanor/Elizabeth?
Colin