Identification should be transparent to the pilot - ie ATC don't discriminate between SSR-identification or ADS-B-identification. Both are good for five miles separation and some flights will transit between being identified on ADS-B only, on SSR only or on both at the same time with no change in the service you get from ATC.
There are a lot of areas around the place (namely near the ADS-B receivers) where there's ADS-B coverage virtually down to the ground but no SSR coverage. If you're in one of those areas your incorrect/missing SSR code won't show on ATC screens (naturally, because you're outside SSR coverage) but because there is a matching ADS-B return it will couple correctly to your flightplan. If you do subsequently enter SSR coverage the controller will then see two returns flying along at the same place at the same height, your correctly coupled ADS-B one and a radar track showing 2000 or 1200 or whatever incorrect code you may be sqwarking. That's where you'll get 'sqwark xxxx' to correct it and as soon as the system picks up the allocated code it will all couple correctly.
Having said that, we're *supposed* to issued a code every time but occasionally ATC or the pilot forgets/mucks it up. It'll become apparent as soon as you do enter SSR coverage though and there's usually no harm done.
(It works the other way around as well, when departing from an aerodrome inside radar coverage the SSR code might couple correctly but the ADS-B Flight ID might still be showing the last flight number, for example)
SSR codes won't go anywhere anytime soon. The existing enroute SSR network will remain in use particularly around the J curve for some time yet - AsA is in the middle of a significant replacement project of the radars around the country which will keep them operating for another twenty years or so at least.