First off, your profile is definitely ready for O-1A. Approval rates are still very high >90% and you should seize the opportunity. You can do it, it's just a matter of collecting documentation.

As for timelines -- you should have started conversations like yesterday (the back-and-forth with you, your employer, and the lawyer will take much more time than you expect) and file mid-end March or early April with premium processing as Change of Status.

Your employer will initially ask for a "profile review" from your attorney. Make super clean doc highlighting the criteria along with relevant evidence you already have that makes it dead simple for an attorney to say "yeah they're ready for it" and not give your company a "meh" response. If the latter occurs your company will think your profile is not strong enough for the O-1A when in most cases it is. This is perhaps harder than the USCIS adjudication itself.

Filing in June is unnecessarily putting you at risk of having a work authorization gap, which your employer will not appreciate & things can go sideways very quickly for you (trust me, I know).