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[*]All provinces - UK DVLA Driving History
Before you leave the UK get a copy of your DVLA driving history/entitlement and bring it with you. In the UK it costs about GBP5.00 to obtain through DVLA.
Here in BC, when we swapped driving licences, ICBC wanted the original DVLA printout (and kept it) as proof of full driving history.
I would not rely solely on the paper counterpart as proof of driving history.
.[*]BC - 'No at fault accidents' proof.
Make sure your no at fault accident paperwork is on letter headed paper with full company contact information on it. Here in BC private insurance companies seem not too bothered about this, but ICBC appear to be an extension of CIC when it comes to bureaucracy for crossing the t's and dotting the i's!
Check that the 'No at fault accidents' statement shows 'from and to' insurance dates and that your cover (between different policies) is contiguous. They do not like gaps in cover. A couple months gap (over different vehicles) is OK it seems, but much over that and they may not count some periods.
In BC you are looking for 8 years of no claims contiguous proof to get maximum discount. If you can't prove 8 years you will be credited with whatever you can prove on a sliding scale. See below.

-1=5%
-2=10%
-3=15%
-4=20%
-5=25%
-6=30%
-7=35%
-7=40%

The certificate(s) you present should not simply be a 'No claims' cert.
If it is it will be thrown out by ICBC as not acceptable proof of 'no at fault accidents'.
No claims discount can be insured against in the UK and ICBC know that.
Your no at fault accidents/claims cert/letter must say… 'No at fault accidents' 'No accidents' (or wording to that effect).
Ours (translated from Portuguese by an ICBC approved translator at $40.00 a page I might add!) met with what was required, but they went over it with a fine tooth comb seemingly looking for possible errors by way of our documents not meeting with strict ICBC proof of 'Safe Driving Discount' requirements. So, I can see potential problems for others following in our footsteps which is why I have written so much detail on it.

.[*]BC - Vehicle Insurance costs.
ICBC run a provincial insurance scheme and you are required by BC law to carry vehicle insurance to a minimum level through ICBC (A bit like Third part insurance in the UK as being a legal minimum). This can be done through a local ICBC approved insurance agent. These agents also offer other vehicle insurance cover.
You do not need to carry all your insurance through ICBC. You can take out the legal minimum ICBC cover and save a good bit by using private insurance cover to beef up liability, personal and passenger injury etc. A bit like taking Third party only in the UK (through one company) and then going to another company to cover you and your vehicle in a comprehensive fashion.
This will generally save you quite a bit, over going with ICBC for everything.
Take your DVLA driver history/entitlement printout with you when you go to insure your car. They may take a copy. They did with ours.
As a side note: When we bought our Jeep we did it through a well known local (large) used car dealer in the area. They had an 'in house' insurance agent (from a local insurance firm) that handled the ICBC vehicle registration, ICBC insurance and additional private insurance all in one hit. It was he that advised they could save us a few hundred dollars over equivalent ICBC insurance rates. We test drove the Jeep the day before, haggled on the price, struck a deal. Next day we bought it, registered it, insured it and drove it out all in a couple of hours. Job done!
The Jeep is a 2008 Wrangler Unlimited (4 door). 3.8Ltr V6 engine. Automatic box. Insurance was about $1500 for the year. Cover was split between ICBC and a private insurance firm. In effect the cover is about the same as UK fully comp cover.
Some might bitch about the price of insurance here (it is what it is, so try not to compare to what you are used to in another country, it's a pointless exercise in my mind) but then again don't hold back telling your friends and family that fuel is about half the price of UK rates at the pumps! That more than makes up (many times over( for the supposed insurance hike!

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