This is going to be a somewhat unhelpful piece of advice, but it may help others with visa issues, something I have to handle once in a while too. While one would not normally advocate using Avios for WT ticketing, this is the sort of example where a traveller should perhaps consider it. If the visa comes through you can cancel the Avios booking if there are cost effective alternatives. The other idea, which very few people seem to know about, is to book via BA Holidays: you can change names on bookings there, that's a legal requirement if paying a deposit, which you can't on normal revenue tickets.

The UuA is refundable for a small fee, you then revert back to the original ticket. It is entirely possible to be non refundable, but it could quite easily be changeable. I'd have a word with the Canadian High Commission / Consulate and find out if they can issue an entry visa, even if it requires a new travel date.

And I am very sorry for the hassle this must be causing you, you've been very restrained in your posting, it must be a very trying for you and your family.