Any insurer will do this, even the mainstream ones. When I insured my Cefiro with Direct Line, I just did an online quote told it I didn't know the reg number but here's the make and model, got the quote, then rang them up and asked to insure it on the chassis number as the car was unregistered.
Technically you're not allowed to drive an unregistered car on the road, except to and from an MOT, but in your case I think you're allowed to drive around with an Irish registered car for the first 6 (12?) months of your residency in the UK so this should be less of an issue. What some of the Jap import people do is book their MOT for the same day they collect the car from the docks, then drive it straight to the MOT station from the docks (stopping at a services to fit a foglight), and then drive it home. Don't forget to park it off road. Personally I had a 3rd part handle this part of the process for me.
Registering it is fairly easy, go down with all the paperwork you need, not sure what that is from within the EU but in my case it was the deregistration/export certificate, evidence that customs fees have been paid, insurance certificate (valid on the day you visit the DVLA) and a valid current MOT, also done on the VIN. You need the registration fee £50 and the money for however much road tax you want to buy
The DVLA will give you a tax disc, a replacement MOT certificate showing the new numberplate and a "certificate of entitlement" which allows you to get a plate made up at any numberplate supplier. You'll need to then phone your insurer and advise them of the new registration number, bolt your new plate to the car, put the tax in the windscreen and you're good to start driving.