Let's try working backward from your Alan Frost.
He doesn't look 54 in the picture taken in Toronto in 1958, I'd say 30 at most, so born between maybe 1918 and 1928.
For one reason or another, that might not be his birth name. I don't think government policy would have been a factor that recently, but it really doesn't affect the search.
So, do you have any DNA links to the Withington, Lancashire, area, where their Alan Davenport Frost was from?
Where did you test the DNA? Do you only have the general region, or do you match specific individuals?
If you have matched specific individuals, can you trace back their families?
Do you match anyone but the first cousin you mentioned? If you have a few good matches, sometimes you can "triangulate" to the common ancestor.
Are you or your cousin a direct male descendant of your Alan Frost? Have you submitted your y-DNA anywhere?
Where have you posted the DNA results? Try uploading them to a few sites, you might find more matches. Ancestry will not accept uploads, but if you tested there you can download the results and upload them to gedmatch or familytreeDNA, both of which have helped me and someone else I know. I think myheritage.com also accepts uploads.
Is the match in Rio helpful? Have you talked to them? Even if not, can you work backward from them?
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added: did you know this first cousin before hand? if not, how do you connect to him?
Your Alan Frost is not that far back. If he had children, brothers, sisters, or 1st cousins, you should be able to find something at least.