Well you can start a corporation in singapore, panama, labuan or others. All foreign income not repatriated to singapore (i.e, it goes in a bank account in HK instead of singapore) is tax free. Panama corporations do have any tax in all income not derived from Panama. Labuan (in Malaysia) has an other low tax structure.
To avoid UK (and Canadian) residents doing a run around on their tax system with these corporations, any corporation who's 'top management' is majority run by residents in the UK (or in the UK) is taxed as if it is a UK corporation. Since your going to be away from the UK, you can remove your UK resident status (you have to do this properly, they can get you with things like still being registered to the NHS or having a residence in the UK still that isn't purely investment property, etc, etc). Once you loose your UK residence (you still stay as a UK citizen forever, your just not resident) your corporation will not have that tax status in the UK, and you do not have to pay taxes to the UK for your worldwide personal income. I don't know how it works when you move around in the EU, it can be different.
You can then travel the world as a resident of nowhere and be mostly tax free, with the majority of your income staying in your corporation, and drawing a living expenses income from the corporation. How you want to structure that income (being a corporate expense or personal income), and report it to the country your currently in depends on you and what your doing. It being an online business of you and your laptop, practically you'll not get any problems from working online on whatever tourist visa your using, since it's virtually undetectable and you don't really have to interact with their economy locally. Especially if it is only for one or two visa stays and a developing country. How you deal with that is up to you.
If you are going to plan which countries your going to, getting a tourist visa before you leave at an embassy in the UK will be useful, since you can get extra long visa stays vs the stamps you get arriving in the airport. For example you could get a triple entry tourist visa from thailand for free which would get you 3x2month stays in thailand. Your not going to get that from a embassy anywhere near thailand, but somewhere far like the UK would be a lot more likely.
For health insurance you can try to get an annual international travel insurance policy if you want to be covered.
I am not a lawyer and you should do your own research, but I hope this helps.