If a driver involved in an accident with you is unable to provide his/her insurance details at the time, then you do have to consider whether or not you believe that driver is, in fact, insured. If the accident is such that the police attend, then they have the facility to check that on the spot.
If you have to go your separate ways without the insurance details, then it is advisable to make a note of everything you can, including full description of vehicle and driver. Do not accept any "I am in a hurry" excuses.
Later, although technically naughty, you can do an insurance check via your computer.
On the website
www.askmid.com you can enter, quite legally, your registration number to check if it is recorded as being insured. This is a legal obligation on the part of your insurance company, and you are legally obliged to tell them if it isn't there.
Now comes the naughty bit. You put the other person's reg. number in the box, and you have to tick another box to confirm you are the owner/keeper, etc. There is a warning that you are liable to prosecution if you are unable to show good reason for this action. But, under the law, you are entitled to that person's insurance details, therefore you are entitled to know if he/she is insured, so how can you not have good reason?
If push came to shove, I would do it.