What were the godparent requirements back in the 1960s? My mother was baptised in the Catholic Church in the 1960s. 

I received information that her godfather (her uncle on her father's side) might not have been Catholic but rather Anglican. Could an Anglican stand as godfather in a Catholic Church? Has this practice changed over the years?
TL;DR: An Anglican is neither then nor today permitted to be godparent at a Catholic baptism.

The then actual Code of Canon Law 1917 (Codex Iuris Canonici, CIC) says:

Can. 765 CIC/1917 Ut quis sit patrinus, oportet:
[...]
2° Ad nullam pertineat haereticam aut schismaticam sectam, [...];
(my translation)
can. 765 In order to be godparent, it is required:
2° He belongs to no heretical or schismatical sect, [...];

In the former terminology the Church of England is a heretical and schismatical sect (we now use other terminology). So an Anglican were not able to be godparent in a Catholic baptism.
This requirement did not change with the reform of the Code 1983:

can. 874 § 2 CIC/1983 A baptized person who belongs to a non-Catholic ecclesial community is not to participate except together with a Catholic sponsor and then only as a witness of the baptism.

The reason for this is also given:

can.  872 CIC/1983 Insofar as possible, a person to be baptized is to be given a sponsor who assists an adult in Christian initiation or together with the parents presents an infant for baptism. A sponsor also helps the baptized person to lead a Christian life in keeping with baptism and to fulfill faithfully the obligations inherent in it.

This function cannot be fulfilled by someone who does not hold the whole Catholic faith and is not in full communion with the Church.
By the way: This rule cannot guarantee that there were and are no Anglican godparents in reality. The priest should check this, but maybe not everyone does this or he just ignores the rule.