Your first port of call should be your insurance company who will guide you through the process and tell you what is required. Presumably you have winter sports insurance.
You should get copies of every expense and every form - the mountain rescue should have given you a "rescue" form; if he was transported in an ambulance there should be a form from them; if there is anything to pay at the hospital, again the hospital should provide copies of any bills. Your GHIC card should cover any medical treatment, but for example in Austria it is common to pay for hospital meals. Medicines from the chemist (apothek) should also be covered by GHIC, or should be refunded by the NHS. You can call NHS Business Services if you need more information about what is covered under the GHIC card.
With regards to compensation etc.
You should be able to claim a refund on your husband's lift ticket. Go to the ticket office with a copy of the discharge note from the hospital (assuming he has been discharged) or confirmation of the injury from the hospital.
Your insurance should cover any transport home (if he can't travel on the booked flight) PLUS any transport in the UK - for example if neither of you can drive your car home from the airport or he cannot travel in that car.
Your insurance should cover any physiotherapy etc. that he might need.
More serious stuff:
@steelseries " Its not like where you have a car accident - you don’t need to exchange details and establish fault to get the medical bills paid."
Correct, but it is a legal requirement in Austria to stop and exchange details if you are involved in a skiing accident. For medical bills, the OP's husband should be covered by the GHIC card for immediate treatment, but for ongoing treatment the OP's insurance company would look to reclaim any costs from the other party (via that person's insurance company).
"I’m not sure if you’re after anything else but if you’re talking about (other compensation) then that’s a civil thing (between the parties) and presumably you’d have to sue him (for what? Some sort of Negligence?) in the jurisdiction in which the accident occurred (so, Austria) which is obviously completely impracticable."
The key phrase here is "Liability Insurance" which skiers should have. UK travel policies include this, as do most household policies in Europe and this is also covered in Europe by membership of many sports clubs - e.g. the Austrian Alpine Club.
To take an extreme example to illustrate - imagine that the OP's husband is a self-employed bricklayer and semi-pro footballer. If he cannot work for 3 months following the crash, he might sue the other party for loss of earnings. If he was on the brink of a full-time pro contract as a football player, if the skiing injury ended his career he might sue for loss of enjoyment and potential loss of football earnings.
His winter sports insurance should (hopefully) include legal fees cover, and it then becomes a matter for his insurance company to cover the costs of the civil case - in effect his insurance company makes a claim against the insurance company representing the other party.