Hi
I went to have a look for the baptism of Jane Middlemiss today but unfortunately had no luck as I had not realised the records I was going to look at only went up to 1812.
The records for Branxton do exist but it would involve going to Northumberland Archives, Newcastle or Durham as explained on the Genuki page for Branxton.
http://www.genuki.bpears.org.uk/NBL/Branxton/Another problem you will have is all the places you are interested in are very close to the Scottish border and there is a large non conformist population so not finding say a baptism in the Anglican records does not mean there is not one, as it may be in a non conformist register somewhere.
I did look at an index for Branxton from 1813 to 1839. There were no baptisms for a Middlemiss (but as this is only an index it could be Jane was missed). There is a marriage in 1824 for an Andrew Middlemiss – possibly a brother to Jane and Elizabeth. He married Elspeth Tarbit and this was a dissenter marriage. There are some Middlemiss burials but none that I could tie in to your Middlemiss’
What follows is conjecture but may help as a suggestion as a way forward.From the 1841 census we see Jane and Elizabeth (sisters) are living with a William and Margaret Middlemiss aged 73 and 68 respectively. This suggests that William was born c1768 at Norham (from 1851 census) and Margeret was born c1773 at Tweedmouth (from 1851 census). Also with them is a Margaert Hay aged 91 (born c1750) who I suspect is Margaret the youngers mother. Margaret Hay was born out of county according to the 1841 census.
Looking at Tweedmouth marriages there was a marriage of a George Hay to a Margaret Nesbit on 31 Aug 1771 (familysearch says 1772). This Margaret could be the 91 year old Margaret in 1841.
There was then an earlier baptism of a George Hay son of George Hay of East Ord on 8 Apr 1749.
Then there was a marriage of a George Hay to a Frances Stamford on 4 Aug 1745.
Before that the Hays do not appear to come from Tweedmouth, and neither does Margaret Nesbit (later Hays)
From the GRO indexes the death dates of William Margaret and Margaret are
Margaret Hays 1842
Margaret Middlemiss 1855 (or 1860)
William Middlemiss 1853
All this would have to be verified to your satisfaction but I hope it gives some useful pointers