I believe you might be able to live your life without becoming a tax resident under any country's laws, though you might want to look carefully at the tax laws of the countries you are leaving. I don't know about the UK or Poland but Canada, for example, makes it a little tough to become a non-resident for tax purposes even if you aren't physically present in the country if you haven't actually established a residence in some other country.
Besides this, however, what matters more is what you haven't said, which is how are you going to earn an income, where will it come from and how will you be paid? The country where you earn your income may want to tax it even if you aren't physically present, or if not then the country where you are paid (e.g. whose bank the payment is processed through) may be interested in you. Generally you don't avoid taxes by avoiding establishing a residence anywhere, you do so by establishing a tax presence (personal or corporate) in a country that doesn't tax the kind of income you have.
With respect to Mexico in particular, note that if you enter Mexico as a visitor you may not legally do any work there, they try hard to be strict about this and their definition of "work" is very broad. I became an officer of a Mexican charitable foundation and had to become a Mexican resident with permission to do that particular "work" even though it was occasional and unpaid. I'm also not sure a visitor can open a bank account any more; HSBC opened one for me as a visitor quite a few years ago based on my having a US HSBC account, but the Mexican bank later got a US fine for money laundering so I think they're all way more strict now. Should you wish to become a Mexican resident for your three months there (not entirely impossible, I guess, and my experience with Mexican immigration is that they are quite friendly and helpful) be aware they will want to tax you not only on Mexico-source income but also your global income unless you can show you also have a residence elsewhere and make a majority of your income there (with other complexities), which requires that you have a home somewhere else, which is apparently what you are trying not to have. And if you enter as a visitor you'll be counting on them to assume you have a home somewhere else too, since if they knew you didn't they might suspect you weren't coming as a visitor. That said, if you enter as a visitor, work remotely for someone outside the country and have a way to access your earnings for living expenses, they may not notice. Just don't go boasting that this is what you're doing, if it is what you are doing.
I can see why what you seem to want to do might be interesting and fun, but I don't think you should expect it to be tax-efficient as well; just limiting your time in any country doesn't somehow immunize you from paying taxes. It also doesn't seem very secure. No country needs to allow you to enter, if you are entering as a visitor and are asked where you live you may not do well if you tell the exact truth, but telling an immigration officer an untruth can sometimes be the start of a really unpleasant experience. Good luck!
