Most Goalies are not NHL ready until 23 years old so that’s a minimum of 5 years. His KHL contract is until 2022 so this year and next is almost impossible (especially adding in Covid protocols). No reason to rush him anyway as he would not help this team and would do more harm than good.
I think you can only take it year by year and see how things go. But if you want to keep a number in the back of your mind as a placeholder, then 5 years is a good number to use.
Meanwhile, we'll just see how it goes. He has done well this year in the KHL, struggled in WJC, he'll have the same set of opportunities next year for certain. Is it possible he suddenly puts it all together and is miraculously ready for the 2022-23 NHL season? Well, it has happened a few times with other top goalie prospects, so it's not technically impossible. But you would not want to count on that. It's more likely he then takes another 1 or 2 or even 3 years as a starter in some combination of the KHL and AHL before he's ready. 2026-27 would not be an unreasonable season to pin hopes on, therefore.
Hence, if after all the dust settles on this season, and our goaltending continues to be a questionmark, you'd be better off shopping the UFA market for a 3-year contract on an established NHL goalie to tide you over. Lots of goalies are headed for free agency. Will they want to come to Nashville if we continue to have a disastrous season here? Uh, that's a fair question. But one thing we have that other teams do not is cap space. We can pay that goalie far more in a depressed market than most other teams can. Rask, Andersen, Binnington, Grubbauer, Mrazek, Reimer... there are over a dozen NHL goalies headed to UFA this summer. Finding a stop-gap to tide us over until Askarov is ready won't be as tough as it is most years, if that turns out to be what we need to do.