I want to emigrate from my country to U.S., Canada, UK or Australia.
Living in EU, move to UK isn't a problem legally talking, but I can't say the same about the others countries. Basically, you can't move there unless you be sponsored.
Should I say in the cover letter that I'll require sponsorship? Knowing that this will practically knock me out of the selection process. Or is it better to wait and see whether or not I get an interview, and then say it personally?
When applying for jobs in other countries, I usually put something like this in my resume:
Citizen of Sealand, Work Permission for EU, US, Japan
Many companies will state in the job description, "applicants must be current residents of Country X" which is a fancy way of saying, "if you're not allowed to live here now, we're not going to help you live here." Ignoring this (or failing to provide information allowing them to determine if you meet that criteria) will likely upset them as it's a waste of their time (it is a huge burden to try to get someone work permission).
My suggestion is to clearly state what your work status is in the country you are applying to, because this isn't something you can sneak through -- if you want the job you will need to broach the subject eventually, and better to do it early to prevent yourself from wasting time interviewing for a job that won't sponsor you.
I can speak for the U.S.
Most states require a certain number of rejections in an interview process before applying for an H1B (permanent work visa).
Companies in the U.S. also are not allowed to discriminate against workers based on citizenship.
To put it bluntly, it's more work than most companies want to do.
However, if you can find a company that REALLY wants you, they will usually write up a job description specifically for you, wait the apportioned amount of time/number of applicants, and hire you.
I would recommend putting your citizenship in your resume or cover letter, as that will prevent companies that are truly not interested from sponsoring you, and give a heads up to companies that would be willing.
Your resume or CV probably indicates your current location. Usually, this is either by listing your current position or your current university. If the job ad doesn't specify that they require local candidates (or candidates who are already currently legally employable in that country), then I think that the location listed in your resume is sufficient when you are applying for a position. If your location is a concern for the company, the recruiter (or hiring manager, if there isn't a recruiter involved) will ask you about it.
For me as someone who conducts interviews, anything that is related to immigration is handled through Human Resources. If you ask me about it during a telephone interview or an in-person interview, I won't be able to answer any questions about it, and I'll refer you to the recruiter who set up the interview. This isn't because I don't want to answer the question, but because I have no insight into the process.
Note: my answer is based on my experience in being involved with hiring for a couple of large US employers in the software industry, although I do have some experience of being a candidate and getting hired in the UK and Australia as well. My conversations with my Australian employer gave me the impression that the process was similar enough. They've each said that it was quite clear from my resume that I'm an American citizen and they assumed when they got my resume that I would require a visa.
