been in almost exactly the same position as you although in my case it was for £1,000.
Insurers when it comes to property damage hope you will go away, it is a civil matter rather than criminal so the police generally don't want to know
the insurers have a statutory responsibility for liability to third parties - I can find the reference in the road traffic act if you want. however, here the approach I used
1) obtain a quotation for repair works for the damage and submit it to them
2) chase in 2-3 weeks if you have not received an answer ( you won't)
3) chase again saying that if you don't hear, you'll consider taking legal action
4) issue a final demand letter giving them 14 days or legal action will commence
the issue with the legal action is that if it is under £10k you cannot use small claims and that does mean solicitor. If less than 10k run the claim yourself via small claims
I'd expect the insurers to cough up at point 4) as they then know you are serious
don't deal with your house insurers, its not their issue, deal with the lorry drivers.
and yes, when it comes to managing claims insurers are little sh1ts. fight for every penny and do not give in.
they rely most of the time on people trusting them or the old BS about their policy blah blah blah which is irrelevant as it is only civil law that matters
good luck