Short answer is: they don't.
Seldomly games do get long enough to scale, but so far, in both S9 and for most of S10 my games end at around 20 minutes due to FF or by 25 minutes if we don't. About 1/10 go beyond 30.
You can see most champions that used to scale build new things that emphasize early games. For example, Cassio takes Conq to get huge AP and healing. Anivia opts out of mana items (or only buys one) since she takes too long otherwise. Kassadin takes PoM to activate his max-mana ulti plays as early as level 11.
Other traditionally scaling champions, like Ziggs just suffer. There isn't much they can do, since champions achieve lethal status far earlier than he does. His AoE damage and turret-taking capability is also just not fast enough to matter much and his winrate suffers. Either Riot lets them drown or they must buff them as we can see them buffing Ziggs next patch in 10.2. You can see this with Kayle and moving her power breakpoints to earlier Lvl 6 and 11 since rarely does she even get to 16.
Unfortunately, even when games do get to 30m+ its typically better to have had impact in the early game as a scaling champion to have secured dragons and rifts than waiting. The truth of the matter is even if you played a champion that straight up had an ability to destroy the nexus if the game hit 35m, you would still lose most (only 46% winrate) of your games if you're playing to try to stall or reach that point.
Supporting Evidence:
S10 games that end 30m or less: 54% *most fast-paced games
S9 games that end 30m or less: 52%
S8 games the end 30m or less: 50%
S7 games the end 30m or less: 37% *slowest paced games *had games that could regularly last 60m+
S6 games that end 30m or less: 46% *had games that could regularly last 60m+
+Another interesting fact: Games in season 10 end 8% of the time at 20m, where only 1% of games ever ended at 20m in S7 (the slowest season). This implies, that games not only don't last nearly as long as before, that also a pretty heft amount of games are decided almost entirely by laning phase alone, which greatly favors hyper-aggro and base-damage champions/mechanics.