Tyrion Lannister drove from Meereen to Oldtown, then from Oldtown to Lannisport, and finally from Lannisport to Winterfell, always taking the shortest paths. He always drove at a speed
10 km/hr below the maximum speed limits for the stretches he took.
What was his total driving time closest to?

We are told that a car always travelling at the maximum permissible speed, and always taking the shortest route takes 1 hour to reach Oldtown from Meereen, 2 hours to reach Gulltown from Oldtown, and 45 minutes to reach Winterfell from Gulltown.

So, distance between Oldtown and Meereen is 100*1 = 100km

Distance between Gulltown and Oldtown is 25*2  = 50km.

Distance between Winterfell and Gulltown. is 120*3/4 = 90km.

Let the distance between Meereen and Winterfall be x.

Using the pythagoras theorem, $$(100+50)^2=90^2+x^2$$

x = 120 km

For convience, we are refering the vertices as the starting letters of towns.

Using the similar triangle rule, MOL is similar to MGW.

So, $$\frac{MO}{ML}=\frac{MG}{MW}$$

$$\frac{100}{ML}=\frac{150}{120}$$

ML = 80 km.

LW = 40 km.

$$\frac{MO}{LO}=\frac{MG}{WG}$$

$$\frac{100}{LO}=\frac{150}{90}$$

LO = 60 km

The path Tyrion Lannister took is MO, OL, LW.

Distances of these path are  100, 60, 40. The respective speeds are 90, 10, 110.

So, the total time taken will be $$\frac{100}{90}+\frac{60}{10}+\frac{40}{110}\ =\ 1.11+6+0.36\ \ =7.47$$

0.47 of an hour is 28 mins.

So, answer will be 7 hr 28 mins