How can I speed up the process to reclassify to 25V when my Retention NCO says it's available, but no seats are available?

No, there is nothing that you, your CO, BC, BDE CDR or CG can do to "speed up the process" or make a seat appear in an MOS school.
MOS seat priority (99%) goes to IET.
The rest go to reclass contracts.
You contract for reclass, and you are guaranteed to get the first available seat that they can get for you.

Your Retention NCO worded it incorrectly. It's "available" meaning the job may need people to fill it. But no Active Duty job is available if their isn't a class seat. If there is no class seat you are not reclassing into it. As SGM (Join to see) very accurately pointed out, the majority (like 90%) of open class seats go to IET and the remainder go to reclass. Of those reclass, the Active Duty and USAR/ARNG have to split them up.

You're probably never going to go into MOS 25V in the Active Duty. If you do, you probably won't make it past E5. If you find a spot to join the Reserve Component as a 25V, you will probably attend training 2-3 years from now. That's just the brutal truth of it all. Pick a different job, or pick several different jobs you are willing to take

SPC (Join to see) Ask when one may come available... he/she should be able to research that.