I have a baptism for Anne Allom in 1645 but with only father John shown.
I am trying Anc search *ll*m and switch last to n. Hopefully that covers Hallam Allum etc
but also picks up Billingham
Nothing clear in brixworth but these look likely
John Allam Marriage Date: 5 Jun 1623 Irthlingborough, Northamptonshire, England Spouse: Alice Buckmaster
Martha Allam [Matthia Allam] Baptism Date: 13 Aug 1626 Irthlingborough, Northamptonshire, England Father: John Allam
But not finding anything else for Alice or Martha.... so hard to say if this is John Anne's father ..
What I have found so far
http://freepages.genealogy.rootsweb.ancestry.com/~lankshear/Eagle/AllomAnne_JohnCook_marriageBrixworth.htmi did find John Cook of 1687 they had wrong year 1686 and name Coos
And also John Cooke 1644 very likely 3 baptisms on 7 April..