Hm, I "solved" this problem by connecting to the server before qualifying only to download skins, in previous seasons I sometimes did that like 10-15 mins before the race, but this can't be solution for all 150 drivers.
One nice way would be to download a replay of the last qualifying or even start of the race that someone usually posts and get skins from there well before going into the race. Only some helmet skins are left to be downloaded then which takes very short time.