Hi littleliz,
I believe in the case of illegitimate births the father's name could only be recorded on the birth certificate if he agreed to be named as the father and attended the registrars to give his consent.
Of course it could be that although the children had to be registered as McTier ( father not attending!) , they were perhaps also known by their fathers' names -Williamson and McBean?? At the time of Census Agnes or the Householder at "Formerly Hospital" would give the enumerator the names they knew/called the children by. Do you know what kind of establishment "Formerly Hospital" was and what age was Agnes giving on the Census?
You think you have found Agnes on the 1881 Census in Preston - was she married by then or still using McTier?
I'm sorry I've no clue whether children were fostered during these years. Maybe some other Rootschatter could point you in the right direction with that one. I think there possibly were cases where youngsters from children's homes would be sent to rural communities to live within farming families and be an extra pair of (unpaid) hands on the farm. But as I said I'm no expert on that matter.
It could be that by 1881 Alexander who would be 15/16 years old was already out in the world on his own and working somewhere. Or could the children be with grandparents - they might be under McTier again?
All just suggestions worth considering
Looby