I do love a good search but I admit defeat I cannot find him in 1891 even with your info or in 1901 none of the details add up.
I am not sure if the family in Liverpool are not a completely different group. If your John Francis was born in Chelsea and appears to have stayed living in London from 1881 onwards why would the rest of the family have lived in Liverpool between 1851 and 1871 and why is he not with the family in Liverpool in 1871 when he would have been only 14.
I can see that the family got around a bit looking at chidrens birthplaces from 1851 (presumably dut to dads job) so it is conceivable that they were in Chelsea when John was born but why would he not stay in Liverpool with them.

I cannot find Richard (born c1840) anywhere beyond the 71 census, nor Patrick although he could have died. I also wondered if MA daughter age 21 in 1851 census was mistranscribed as Mary Ann wife age 40 Middlesex in 1871 or is it a coincidence that they have same initials and place of birth

. I cannot find son E age 33 anywhere either
On the other hand I cannot find John Francis anywhere before or after 1881 (although you have them in 1891)
Also there is a John Francis McCormack marrying Elizabeth Owens in 1880 Sep qtr at St Michaels Church West Derby Liverpool. So there must be another John F around as the name and place of marriage are at odds with the 81 info.
What a conundrum.....
I think a request for a look on 1861 in lancashire may sort out if John is the son who belongs there (I have the census but no index). Also a look up at Liverpool records for the marriage of John and Elizabeth just to rule them out (Florida Girl often does look ups there although I may be going back in a few weeks time).
I hate it when I can't find people

Donna