Hello
For Catherine Osborne assuming her daughter Ann was illegitimate she started life as Catherine Unwin.
So this may be her at Southampton St Lawrence
Catherine Unwin 13 Nov 1774 daughter of Isaac and Hannah Unwin (bap privately 16th May).
Brother John Brown Unwin 12.11.1769,Sister Hannah 26 May 1771 and Isaac (bapt 3 Feb 1771 privately),Martha Matilda 20 Dec 1772(bap 31.10 privately),and William 31.12.1777(bapt privately 16.6.1775.
They must have had a lot of premature babies.
I can't see a baptism for Ann or Catherine's marriage to a Mr Osborne.
But there are children for William and Catherine Osborne at Southampton St Michael (bapt recorded at Hambledon for some of them but included in St Michaels registers).
Catherine 4 Dec 1792 at Hambledon,Mary Anne 6.10.1794 at Southampton,Ann Elizabeth 18 .6.1797 ,Elizabeth 22.8.1798,Hannah 11.8.1800,and Mary 24.7.1804 at Portsea Hampshire.
So everything would fit.
Sorry
Ciderdrinker