Have been doing some more digging. Charles Taylor's given occupation doesn't make much sense as I read it and I now think it could be a badly written Cotton Weaver.
Have found a candidate that fits this bill - find him in 1841 and 1851, born and living in Hamilton too, but by 1861 he has vanished and cannot find a death for him. Could have emigrated - who knows?
1841 - Young Street, Hamilton
Alexr. Taylor 50 Cotton H.L.W. (H.L.W. stands for hand loom weaver)
Bertha Taylor 20 Tambourer
Matthew Taylor 20 Cotton H.L.W.
John Taylor 15 ditto
Charles Taylor 12
Jean Taylor 10
Alexr. Taylor 7
William Taylor 5
1851 Census - 15 Young Street, Hamilton
Alexr. Taylor 60
Elizabeth Taylor 33
Charles Taylor 22 Weaver
Alexr. Taylor 17
William Taylor 15 - all born Hamilton
The Scottish naming tradition is quite helpful when identifying families. If you google you'll find out how it goes - suffice to say 1st son is usually named after their paternal grandfather and first daughter is usually named after maternal grandmother (which is why my own mother was given the 2 names Margaret Ingram and she had 3 cousins that had these Christian names too).
So, although not married, it makes sense that if this Charles Taylor is your man (right age, right area, right profession) then his first child would be named Alexander and Agnes Coopers eldest son on 1861 census is Alexander Taylor.
At best this is only a theory but seems a pretty darned good one to me.
Agnes Cooper eventually married (as a single woman) to a William Fleming (quite a bit older than her) 1/1/1863 Hamilton. They had a son Robert b.6/7/1864 High Church, Glasgow.
Hadn't previously found William Taylor in 1871 Census but have now found the family - couldn't find on Ancestry but found on FindMyPast.
1871 Census - 30 Sylefield Street, St. John, Glasgow
Wm. Fleming 55 Wool Weaver born Hamilton
Agnes Fleming 42 born Hamilton
Alexander Tylen (?) son in law 18 born Hamilton
Wm. Fleming (?) 13 born Hamilton
Robert Fleming 6 born Glasgow
Can only view transcriptions of Scottish census so quite what is written on the original I don't know. Presumably 'Tylen' is actually Taylor but whether William is actually shown as Fleming
have no idea. However, this is clearly the right family group.
Agnes Fleming died 1892 in Shettleston - informant son Robert Fleming.
Hope this helps.
Annette