In general I haven't had very good luck rescuing fabric with rust issues. Sometimes no major problems, but mostly I end up with some holes.
When I collected vintage fabric I carefully bought brass safety pins and acid free labels that I would write the yardage and other information on. Stored them in a box lined with acid free tissue paper, etc.
A lot of my vintage fabrics had stapled on tags (which rust), pins (which rust), and safety pins (which rust). When they are solidly encased in the rust I have been able to bend/cut them out, but usually can see the flawed areas and so I cut around them when I use them. Since I was selling I left them in place and noted it on my tag, some of the rusted items had provenance to the piece, like a store tag and date on it.
In the case of finished blocks, I'd probably cry a lot and then remind myself that it was "Just a UFO and if it was important to me it would have already finished". Then I'd work at breaking out those pieces as best I can. You can apply a dot of fusible interfacing on the back if the threads break. Finish it the best you can, and then move forward -- with brass pins!