In the 1930's-40's could a father run off with his child?

[TPH]

Thanks, so does that mean that anyone who was under the age of 18-19 at the time would have their records withheld?

[KGarrad]

No.

It means that anyone who is today less than 100 years old, for which there is no proof of death, will be redacted.

FindMyPast update their database monthly (I believe).
Ancestry update theirs once per year.

[Guy Etchells]

I am exploring a complicated part of my family history and there is evidence to suggest the during the late 1930's early 1940's a father could have run off or disappeared with his child.

I'm just wondering how feasible this would be and whether legally this would have been allowed?

Yes it could and has happened.
I can give you an example from a few years earlier, 1899, where an estranged husband had his daughter "kidnapped" from his wife to try to force her to divorce him.
A newspaper report may be seen on Findmypast here (subscription site)

See:- http://www.rootschat.com/links/01qg7/

Basically after infidelity of the husband the couple split up with the wife and daughter going to live with her parents. She would not give him a divorce so he used this to try to force her hand.
After the “kidnap” he was summoned for deserting his wife, but as they were married could not be charged with kidnap. The court granted a separation order, gave custody of the child to the applicant (wife) and ordered the husband to pay £1 per week for upkeep of the child.
Cheers
Guy

PS he still did not get his divorce.

[Maiden Stone]

Guardianship Act 1973 gave mother equal rights to father.

See also:
Guardianship of Minors Act 1971
Matrimonial Proceedings (Magistrates' Courts) Acts 1960s
Custody of Children Act 1891
Matrimonial Causes Act 1878
Infant Custody Act 1873
Infant Custody Act 1839

A potted history of child support in Britain with relevant legislation:
www.childsupportanalysis.co.uk/information_and_explanation/world/history_uk.htm

[Maiden Stone]

Based on what I know it is impossible to say whether it was with or without the mother's consent.

It is possible that it maybe due to the mother being unfit.

The mother was granted an Absentia divorce from the father in the mid 1940's. This would suggest there was no contact with the father and that she did not know his whereabouts or that of her child.

If the parents were still married to each other and there hadn't already been a separation order or a custody application by the mother  by the time the father "went off" with their child, then I think the father was acting within the law. That's just my opinion based on my understanding of family law in the first half of 20th century.
 You need to know what the legal situation was at the time you think the father went away with his child. There were changes to divorce laws after both World Wars.