Your neighbour's insurers know a policy will be in place to cover your car so that will be their first port of call when wriggling out of any responsibility.
They'll probably only be interested in talking to you if they think that would be a cheaper route than accepting the risk that your insurers might try to claim their costs from them.
It might be worth asking for a full and final settlement figure from their assessor to see what he says but be very clear that he will be hearing from your insurers if it's not forthcoming. Point out that a hire car and loss of earnings will be involved and that might encourage them to accept a generous settlement rather than get involved in wrangling over costs.
To be honest, given that loss of earnings is something you'll presumably be pursuing, if you have legal cover on your car / home insurance, now would be a good time to get them involved.
If you've got to fall back on your car insurance, do it as soon as you get the feeling that her insurers are fobbing you off. If you've sat on the claim for days and then phone them, it'll look suspicious.