If I live in UK and it is a member of the EU, do I have to buy my travel insurance from a UK insurance company or can I buy it from an insurance company in another EU member country?
The answer for this really depends on what the policy wording is for any insurance provider. In most cases however, only people who have been resident in a country for a certain period are allowed to sign up for insurance policies within the country. In addition to that, if the insurer goes bust then you can only appeal to the financial regulator in a country for compensation if you're resident in that country.
Yes and No!. I currently hold travel insurance from another EU country then where I live, but I also got a previous insurance canceled when I changed country. The cancelation letter was the nicest letter I ever got, that they regretted they informed me that I had been erroneously been insured for some years. The small letters dictated that every journey should originate from the country they were in. They were so nice to reimburse me all the paid premium over the years I was technically uninsured. Luckily nothing happened in that period.
Since I work in one country and live in another and sometimes start a personal holiday right after a business trip (meaning I am not using the return leg of the business trip) things can get dodgy. I found a solution in a foreign credit card company, which offers "free" travel insurance if I opted for their gold card. I say "free", since the annual credit card fees is still quite hefty. Compared to a local travel insurance the annual credit card fee, was only 10 EUR more expensive then the annual premium offered by the insurance company "around the corner". I only took it since it served my need for a insurance that doesn't requires a trip starting in your home country.
My impression is that only Global credit cards offer these insurance's and not the national insurance companies, hence the yes and no answer. At the end of the day, no matter what they say, you always need to check the small print.
The EU Internal market for Services specifically allows, but does not oblige EU insurance companies to sell policies anywhere in the EU. (It's one of the services which are not covered by generic services rules).
However, be aware that such policies may exclude travel within the nation from which they're sold - which would be abroad from your perspective. E.g. buying travel insurance from a French company could exclude travel within France.