Olly, we've just been through a similar situation, ie a non-fault total loss, and I'm sorry to say you're probably at a disadvantage in this situation with only 3rd party f&t insurance. We had our claim for the car negotiated by our insurance company and the advantage was huge - a very swift settlement at full dealer retail value. As Andy has said, I hope you have legal cover as it can be very useful, although I guess there aren't any injuries to claim for.
Did you sign any paperwork for the hire car? Is there a daily rental figure quoted? Is there an invoice address? Is it stipulated who is responsible for the hire charges?
Be aware that there's a legal requirement (not an insurance industry guideline, but a legal requirement) to minimise your expenses following a road traffic accident. This means that of you can use another vehicle, or public transport you should do so. You can claim the cost of this, along with a daily loss of vehicle inconvenience allowance (about £10 a day I think). Delaying sending any paperwork to the insurance company may be seen as not minimising your expenses and will only lengthen the time taken to reach the point of receiving an offer. I would recommend dealing with the insurance company in a fair and responsible manner - you may just be pleasantly surprised with the reasonable way that they treat you.
Good luck with sorting it out, hope it's straightforward.