I think to some degree, they already have movement between “Commonwealth of Nations” members, which also include Pakistan, India, Nigeria, Kenya, etc.
https://en.wikipedia.org/wiki/Commonwealth_of_Nations#Objectives_and_activities
I say this because around 1980 a company I then worked for sought to bring an Indian national to the US, because he was a valued employee.
It was stated at the time he could get to Britain or Canada for sure, if not the US.
So we went ahead, he succeeded to get a US Green Card and eventual citizenship.
Not anymore - yes, we had that until about 1983, but the European Union made Britain stop giving such free movement to other members of the Commonwealth. That’s one of the reasons why some people were anti-EU (although not one of the most important reasons) because it forced the UK to abandon some traditional friends. I was travelling quite regularly between Australia and the UK at the time, and I remember the first time I had to use the ‘Aliens’ gate at Heathrow.
(I have joint UK/Australian citizenship now but at that time I only had Australian citizenship - there had been no point up until then in asserting my right to British citizenship by descent - now there was).