We are planning to shift to Canada. We have a pug named Leo who cannot go by plane as brachycephalic dog breeds are not permitted to travel by most airlines. We cannot leave him behind so the only option is to travel by ship from India to Canada. But we do not know of any way to travel by ship. Can anyone offer guidance on this issue?
Have you looked into in-cabin travel for your pet? A few airlines do allow pets to travel in-cabin on international flights. See this Q&A on this site (which may be a bit out of date by this point), this 2016 guide from dogjaunt.com, and this more recent run-down of airline policies (focusing on North-American-based airlines.)
In-cabin travel is generally considered less risky for brachycephalic breeds than cargo-hold travel, and airlines will allow some breeds in the cabin that aren't allowed in the hold. For example, Air Canada bars brachycephalic breeds from the cargo hold, but has no such restriction for in-cabin travel, and Air France allows it as well (though they note that there is still a risk and encourage you to consult with your veterinarian.)
In general, this will only work if your dog is under a certain weight (pugs are small breeds, but I've known some tubby pugs who wouldn't meet some of these limits) and will fit comfortably in an under-seat carrier. If this might be feasible, and you haven't done so already, I would call as many airlines as you can and inquire about their policies before resorting to the hassle of freighter or cruise ship travel.
Unfortunately it could be very difficult and costly.
Personally I would check with airlines and cargo airlines about solution. Often there are much more possibilities (with extra prices).
Cruise ships usually do not allows pets (but for blind people, and often not in such case: a dog cannot help in case of emergency, so now cruise ships requires an accompanying person).
There is one lines/cruise ship: Queen Elizabeth of Cunard which allow dogs in kennels (you may visit it just for limited time, every day). But this ships do usually UK to US (and back).
Cargo ships are also a possibility, but it would often not be comfortable (across the pacific ocean). Or car + train to UK.
So I think it will be much easier to check possibilities with plane, and look for specialized pet transport companies.