Has your claim been denied by the insurance company or do you just feel like they're taking too long? If the claim is denied, you can certainly appeal it. If it's just taking a while (it happened in mid-May), there's not really much you can do. Claims can take 6-8 weeks to get paid.
On a separate note, was the tree otherwise healthy and some heavy winds knocked it over? Or was it a sick, hollow, or infested tree that should have been taken down long ago? Because if it was the latter, you may be able to sue your neighbor for the remaining damages.
It sounds like you are relying on your neighbor's insurance company to pay, and they may very well do so. Make sure to tell them all your damages and incidental costs. To the extent you haven't notified your own home owner's insurance of the damage -- also passing along the information you have about your neighbor's insurance -- you should. That way any available coverage from them isn't denied as a result of your failure to report the claim promptly. And they might also be helpful dealing with your neighbor's insurance company.
If coverage is denied, you may be able to sue your neighbor or your own insurance company...you'd need a litigator. And of course, you can hire a litigator at any point to help you: someone who has handled insurance litigation before might be helpful.
Best of luck.
This post is for informational purposes only. You should not construe it as a legal opinion on any specific facts or circumstances, and you should not act upon this information without seeking professional counsel. Publication of these materials and your receipt of them does not create an attorney-client relationship.