Do I make up missed travel prayers shortened or in full?

It depends on where you make it up, and scholars differ. If you make up a missed travel prayer while you're still travelling, most scholars say pray it shortened. If you make it up after getting home, the Hanafi and Maliki schools say still shortened, while the Shafi'i and Hanbali schools say pray it in full.

The basic rule

Shortening (qasr) is a gift for travellers. Allah says:

وَإِذَا ضَرَبْتُمْ فِى ٱلْأَرْضِ فَلَيْسَ عَلَيْكُمْ جُنَاحٌ أَن تَقْصُرُوا۟ مِنَ ٱلصَّلَوٰةِ "And when you travel throughout the land, there is no blame upon you for shortening the prayer..." Quran 4:101

Umar was puzzled that shortening continued even in safe times, and the Prophet ﷺ told him: "It is a charity Allah has given you, so accept His charity" (Sahih Muslim 686). See What is qasr prayer?

And a missed prayer is simply prayed when you remember or wake up (Sahih Muslim 684).

Missed while travelling, made up at home

Hanafi and Maliki: pray it as two rak'ahs. The make-up copies the prayer as it was owed, and it was owed as a traveller's prayer. Aisha said prayer was first prescribed as two rak'ahs, and the travel prayer stayed that way (Sahih al-Bukhari 350).
Shafi'i and Hanbali: pray it as four. Shortening is tied to actually being on a journey, so once you're home the concession has ended.

Both views are respected. Follow the school you usually follow or the guidance of your local imam.

Missed while travelling, made up while still travelling

Most scholars, including the Shafi'is in their main view, say you can pray it shortened, since you're still a traveller. This is the easiest case. For how long the traveller status lasts, see How many days can I shorten my prayers when traveling?

Missed at home, made up while travelling

The four schools broadly agree: pray it in full. You owed a four-rak'ah prayer when you missed it at home, so that's what you make up, even if you're on a plane or in a hotel by then.

A quick summary

Travel prayer → made up on the trip: shortened.
Travel prayer → made up at home: shortened (Hanafi, Maliki) or full (Shafi'i, Hanbali).
Home prayer → made up on the trip: full.
Fajr and Maghrib are never shortened.

Key points

Make up missed prayers as soon as you can.
On the trip, a missed travel prayer can be shortened.
At home, the schools differ; both views are valid.
A prayer missed at home is always made up in full.

Tip: The easiest way to avoid this question is to combine prayers while travelling. If you know a long drive or flight is coming, pray Dhuhr and Asr together before you set off. See How do I combine Dhuhr and Asr when traveling?