Hi Dan,
John appears to be in Tavistock in 1851 with Elizabeth (oldest child aged 8 )
1841. I suspected he was in Tavistock unmarried a 20 year old copper miner but family didn't seem right. Then in Mary Tavy I find a John also aged 20 (copper Miner) wife Elizabeth 25 and son William aged 3 months. Appears to be residing in a part of the same house as William 50 (copper miner, not born in Devon and his children including a Susanna) Found Susanna's baptism in Tavistock and her younger brother William's baptism in Mary Tavy listed on the IGI, so this is your John and he is married by 1841, though given his age probably not long married. I would expect the marriage to be in the Tavistock district but alas I am still not spotting it, so may pay to get one of the children's birth certificates for her maiden name and try again once you have that.
All the best
Kris 
Kris,
Yes that is the correct John, his father is William and his mother is Susanna Weeks. William and Susanna were married 21 May 1818.
I have seen a marriage for a John Barkell and ELizabeth Nichols in Plymouth but I suspect this is the other John Barkell that you have mentioned as that marriage took place in 1844, well after the birth of Williiam that is seen in the 1841 census living in the same abode as William born c. 1791.
The young William must have died between 1841 and 1846 because in the 1851 census John is listed as the oldest child and another William is only 5 years old.
Thanks for your time in looking though