I chose the case where she ignored the subpoena as then the get will possibly be written without her present, she’ll be notified that it’s waiting for her to pick it up and he can start the process (you wanted to know when he can start the process and that’s a simpler case).
If she responds to the subpoena and comes to Bais Din and the two of them can not agree on a single thing and decide to fight it out- it’ll get more complicated. If Bais Din sides with him and issues the get but she refuses to accept it- it’ll stay there until she decides to pick it up. Either way- once the get is written he can initiate the process. If Bais Din refuses to write a get maybe he’ll go ahead and find a different Bais Din- he may have to try multiple Batei Dinim until one sides with him. Either way- until a get is written and waiting for her to pick it up I don’t think he can proceed.
I can’t say what the Rabonim had in mind when they decided on the heter as I don’t even know the names of the Rabonim but I’d like to think that they took the following into account.
If the wife is R”L in a coma- it shouldn’t be too hard for any of the 100 Rabonim to verify the facts through witnesses and sign on. If the wife ignores the subpoena and the get is written in absentia- it may be a little harder but the 100 Rabonim should be able to verify the facts through witnesses and sign on. (It can be costly and time consuming if each ones tries contacting the wife to hear her side of the story and have her explain why she didn’t show up to a legitimate Bais Din, but it is doable.) If the husband kept getting told over and over again that he does not deserve a divorce until he was able to find a Bais Din that is willing to write one up – hopefully the 100 Rabonim that he approaches will catch on to what he did and not sign on.
What if he goes to three friends, calls them his Bais Din, has one of them write up a get and obtains permission to start the process (without his wife even knowing about it) and he gets 100 friends to sign on to the heter? This shouldn’t be too different than some guy calling himself a Rav Hamachshir and giving a Hashgocha- if you hold of the Rov you can go ahead and eat what he certifies, if not stay away from the food.
Similarly- If the husband has the certification that the get was written and 100 signatures that say he can go ahead a marry a second wife- if you trust the Bais Din and 100 signatories- let him marry your sister / daughter… If not- stay away.