Hi Olwyn,
Unfortunately the baptisms only go up to 27 December 1812, so if he was born March/April 1813 he won't be there. Even if he was born as early as 1810, sadly there's no baptism for him in the years 1810-12.
The only Middleton entries are:
Edward - witness to the marriage of
Margaret Middleton (of this parish) to Robert Raine of Staindrop [neighbouring village] on 7 December 1774
Elizabeth Baptised 19 May 1776, daughter of George Middleton of Barffot [no idea where this is; doesn't appear to be a neighbouring village so maybe the name of a farm or house within the village; there is a farm at Barford Moor about 2 miles or 3km North of Whorlton]
George - also a witness at same marriage as Edward above.
Jennet Buried 23 July 1688
Margaret married 7 Dec 1774 to Robert Raine.
According to a quick search on FamilySearch there are quite a few nineteenth century Middletons whose place of birth is shown as Whorlton in the censuses. The 1841 census has a Middleton family consisting of 9 individuals spread over 3 generations living at Arlaw Banks, 1 mile from the centre of Whorlton (see centre of
this map). The address comes under the adjacent village of Gainford, but it's physically closer to Whorlton. No Joseph among them but the oldest son John is still living at home, born 1816.
Sorry not to have been of more help.