Why not allow citizens of CANZUK countries to join each other's militaries more easily? For example, a Canadian citizen who undertakes residency in the United Kingdom should be allowed to join the British Army as an Officer or NCM and vice verse or, alternatively, to transfer from the Canadian Army to the British Army more easily and maintain rank and seniority.

We can already do this on a limited capacity, admittedly there is a lot for paperwork and a bit of red tape involved, but it is possible. The red tape is not necessarily the government, but the actual defence force. 

I will try and explain what I mean by that;
Take the Royal New Zealand Navy for example. A small navy with limited number personnel and postings. Limited number of personnel in the various branches. If there are an influx of transfers ie; people wishing to leave the RNZN to say join the RN because they wish to serve on an aircraft carrier then you create a shortage of trained and experienced personnel in the RNZN. 

And vise versa if there is a influx of personal wishing to join the RNZN from other navies and for example there are another 100 cooks/chiefs wanting to transfer from the RAN , RCN, and RN. (Yes this is an enhanced scenario and it would most likely never happen this bad and would not be like that... one would hope.)  There are only so many postings that chief can goto that is already filled in the RNZN. So all of a sudden the RNZN has 100 extra chiefs that can't be posted to a base or ship due to those postings are already filled, and even if it was an even number of of chiefs from the other navies, it is starting to create a shortage in their navies.

There are a limited number of transfers allowed in the RNZN and most likely across the NZDF as a whole, when I was serving their was a Leading Hand Radar specialist that served in the Falklands War and then transferred to the RNZN soon afterwards. He kept his rank and position. There are also a number of officers in various positions that have been able to transfer as well. These were not temporary exchanges but full transfers. But I would surmise that it is limited. If there is a shortage in an area then it would make it easier to get the transfer you wish.

RAN - Royal Australian Navy
RCN - Royal Canadian Navy
RNZN - Royal New Zealand Navy
RN - Royal Navy