Much like other beaurocratic processes, all things take time. When you consider how many parties are involved - you, the other party, your insurance, their insurance, independent claims assessors, potentially police, witnesses (if any).
Each party isn't going to act on it immediately - they have other work today. Maybe they'll get to yours this week, maybe next month. I have a friend who works in city planning, and the size of their team (small) versus the size of applications (large) means there's no way they can get to yours any time in the next six weeks, without someone up the chain saying it gets priority.
So, now you're left in multiple queues for multiple months, unless there's intervention. So what else can you speed up? Time to collect data, and error reduction.
With your claim, include EVERYTHING. Got a photo of the accident? Include it, as the claims assessor or police might want to confirm. It prevents contention as well. Got a written note? Include it. Unsure? Include it.
What this does is reduce the number of inter-department questions. If party A opens your file after a month, needs an answer, they'll message party D, close yours, and it goes back in the pile until the reply comes, and they'll add it to the folder. It's still in the pile though, and has to wait until they're free again. Any information that is readily available, and well indexed is going to help.
Talk to your insurance provider as well. Make sure your details are up to date for the claim, and if there's anything else they need. Get names of people you speak to - it helps for accountability, and finding who knows about your case. Get case numbers, if possible.
Document every call you make, every letter you send/receive, with details and date. This comes back to my earlier point of providing proof, and helps speed up the process.
Finally, and this comes back to proof, but photos are the best visual indicator. If it's just a description of the incident, it's on paper, but a photo is visual and can actually help people remember your case, as well as help with documentation of the situation.
(Example source backing up these points)
Bonus point, and I hesitate to say this, but often the squeaky wheel gets the grease. Call. Find out what's happening. See if you can help speed it up. Ask where it's at. Don't be annoying, but do ask for a date - 'will it be done by March?' or similar. Document these dates. If they don't meet them, ask to speak to a manager about expediting this, as they're now running behind schedule and you're waiting on them. Often people will focus on a case to get rid of it, if it's becoming a source of pain for them ;)
(Disclaimer: this can backfire, depending on your and their approach/mood when this occurs.)