Using the Oxfordshire Family History Society Search Suite, there is a baptism in Witney for a Thomas BIRCH on 13 April 1798, the son of Thomas and Esther. However...
The only other BIRCH/ BURCH record I could find in Witney around that time was the baptism of Maria BIRCH on 26 November 1800, the daughter of John and Esther. I could not find a marriage anywhere in Oxfordshire for a Thomas (or a John) BIRCH/ BURCH to an Esther.
I fear this is not as hopeful as you would have liked.