There was recent news that this train has (or should have?) restarted in April:
https://www.azernews.az/region/190928.htmlAccording to the company, starting from April 1, 2022, train No. 647/648 will start running on the Nukus (Uzbekistan)-Beyneu (Kazakhstan)-Nukus route.
This train will depart from the Nukus station on Tuesdays, Fridays and Sundays, and on Mondays, Wednesdays and Saturdays will return from the Beyneu station.
When I check the Kazakhstan state railway website, it shows this train on the timetable:
https://bilet.railways.kz/sale/default/station/schedule (enter BEINEU as the station and choose a Monday/Wednesday/Saturday for the day)
If I try and book a seat on this train using the same website, I'm told "No vacancies" for the dates over the next couple of weeks I attempted. It might be that tickets are only available locally or possibly the train isn't actually running yet. Either way, the situation looks more hopeful than in 2020/2021 but until tickets start being sold and someone actually sees the train running it won't be 100% certain.
In answer to your question, I'm not sure of alternatives, but now crossing Kazakh/Uzbek border by land no longer requires an IDC permit, hitch-hiking could be an option? The border crossing page on this website has some info, as does the forum topic referenced by it:
https://caravanistan.com/border-crossin ... tan/#tejen