3rd party insurance claim help

Hi guys. I have a bit of a situation.

i need some advice... I made a third party claim against someone who crashed into me. The damage wasn't bad and the car is still drivable.

I was told that a third party claim requires you to get the quotes yourself and do all the running around. Now I am told after taking time off work to get the quotes etc. I have lost income, spent money on petrol and I submitted the quotes etc last month. After loosing my temper end of last week suddenly things are getting done but now they making me take the car to get assessed again.

I understood that third party claims, the person who was making the claim did all the work then they were paid out in cash and then repaired their car in their own time?

Do I have the right to get them to pay me out or do I have to listen to them and do what they want? I don't have the time to send my car in for repairs now and I need the car for a holiday trip next month.

I just want to know if I I can demand they pay me in cash?

I used the search bar but couldn't find a similar situation to mine.
any help would be great. Thanks guys