Getting car insurance in Britain, even if you have a foreign driving licence, is quite possible. The majority of UK insurers will only cover people with UK driving licences but around 40% of drivers on this website will provide quotes for those with non-UL licences.
If you are only staying for a short while and you brought your own car over with you you may well already be covered by your own insurer in your home country, particularly if you have come over from a country within the European Union. This would typically cover you for a maximum of three or perhaps six months, and the usual cover is third-party only, which would meet the legal requirements in the UK. This would mean that, in the event of an accident you were judged to be responsible for, your insurance company would pay for any damage or injuries that you caused to other people, but not to your own car or yourself. Your own insurer may of course be prepared to extend this cover to fully comprehensive.
If you have not brought your own car over with you, it is possible to buy short-term car insurance which will cover you for a maximum of 28 days. However, it is possible to have multiple policies if you need these for a longer period. These tend to work out more expensive than yearly policies, but can be cost-effective if you only need cover for a few days, weeks or the odd month or two. You can get more information here about short-term policies.
You may be able to rent a car over here. The vast majority of car hire companies have insurance policies which can cover people with driving licences from many different countries. It is important that you not only bring your licence with you, but also bring an International Driving Permit, which you can get in the country that your driving licence was issued in.
If you are intending to stay in the UK as a resident you may find it much better to apply for UK licence. There are different rules and regulations on how to do this which would depend upon which country your original licence was issued in. There are also different regulations between Northern Ireland, and the UK mainland. You could get more information about this here.
If you still have problems getting cover, contact a friendly insurance advisor now.