As a general rule of thumb, your insurer (if you were insured at the time) do not work for you, they work for themselves. They would have been likely to try to rip off both the other party's insurer and you.
Claiming through your own insurance, rather than directly through the other party's insurance, is generally only advised when claiming directly is not viable - e.g. the other party's insurer is being overly obstructive, you need an alternative vehicle while yours is being repaired, and cannot afford to hire a car and claim back the cost (or buy another car, sell it afterwards and claim for any costs/losses in doing so if duration of hire period id likely to result in excessive cost - injured party has a duty to take reasonable steps to mitigate any consequent losses he will be seeking to recover.
Favourite tricks include settling 50/50 on your behalf because they CBA to risk going to court and multiplying the true costs through their own claims management company and then pinning the costs on you in the event that the other party's insurer avoids paying them on the grounds that they were not properly and necessarily incurred costs.
Short of resorting to blackmail, the other party's insurer cannot use the fact that you were not insured to leverage any arguable dispute regarding liability if there is no arguable dispute - in other words, if the facts and evidence are clear-cut and undisputable. Not being insured is not a causal factor in the collision (unless you were on the phone trying to get cover when the collision occurred).
The other party's insurer will either be approachable, or they won't. Some initially refuse to deal with individual third parties, whereas others will bend over backwards to avoid the injured third party claiming through their own insurer (with a greatly 'inflated' claim) - once there are satisfied that the liability lies with them.
Technically, your claim is against the other driver, but his insurer is obliged to pay out if you successfully claim against him in court. Insurers generally like to avoid court where possible as it adds to the costs. In practice, in most cases, the injured party claims through their own insurer (mostly due to ignorance) or the guilty party's insurer, and the matter is settled without going to court.
Claiming directly against the other driver is your backstop. Your first port of call is to contact the other party's insurer. They are likely to ask for the details of your insurer. If they think you are being evasive, they can readily check whether there was insurance in place at the material time, but the main point is that you are not claiming through your own insurance, so you, rather than your insurer (if you had one) are the point of contact and the decision maker.
If they won't entertain a claim from an individual, you can either claim against the driver yourself, or instruct a claims management company to do so on your behalf. Claims management companies, and their grossly inflated claims are a large part of the reason that some insurer will bend over backwards to look after injured third parties directly, and also the reason you should be wary and do your homework before instructing one.
This is a bridge to cross if we get there (if the other party's insurer refuses to entertain a direct claim).
The Financial Services Ombudsman enforces some fairly strict rules regarding how your own insurer (if you had one) treats you, but you cannot complain to them about another party's insurer as you are not their customer.
I am not aware of any legal mechanism that would force the other party's insurer to engage with you prior to a court claim (or a Letter Before Action) being issued.
At this point, don't overthink things. Contact the other party's insurer. Understand that if they want to know whether you were insured, it would be trivial for them to find out. Understand that it is not a material/causal fact/factor. Don't go out of your way to tell them something that you are concerned might potentially complicate matters, but most definitely don't lie if directly asked (particularly as the conversation will be recorded "for training purposes"). Understand that if they refuse to entertain a direct claim from the injured party, that merely makes the process more long winded.
It would probably be helpful to get across the strength of your evidence succinctly - you have video of the aftermath, position of the vehicles and damage, single lane exit, changing story 3 times - also witness who saw the collision and prior near misses.
