Hi,
If you check Durham Records Online there are some McConvil(le) burials at Gateshead including a Peter born around 1811 died 1868 abode Leonard's Court. Also a John born around 1860 died 1863 abode 6 Nelson Street.
https://www.durhamrecordsonline.com/NewSearch.phpFamilysearch has a couple of baptisms at St Joseph's Gateshead for a Roseanna McConvill 1862 (died 1863 of Leonard's Court from above DRO site) & 1865 dau of Peter and Catherine Nee SMITH and also a son Jacobus (James?)1859.
1861 census indexed on familysearch as McBoniel
Leonard's Court Gateshead RG09/3803/52/page 33
Peter 50 Ireland labourer in soap manufacturers
Catherine wife 34 Ireland
Peter son 12 born ?
Michael son 5 Gateshead
James son 1 Gateshead
That is a big gap between Peter & Michael - is Catherine a second wife or did some children die?
1871 Catherine a widow which ties in with burial of Peter in 1868, but they have reverted to name of SMITH
Gateshead
Catherine Smith 33? Ireland
Michael son 14 Gateshead
James son 12 Gateshead
Rose Ann dau 6 Gateshead
I have not found Susanna in the 1861 census - she may be working away from home or even visiting family back in Ireland.Her surname may be mistranscribed.
With regards to George Davidson senior - he may have remained in Scotland or moved to England with his son. Try trade directories for the Gateshead/Newcastle area . Census may have name variant Davison so best to check that too.
Hope this helps