There is a lot to be said getting a Family Lawyer involved, I have no idea what would work legally in your Country, now I could guess, but…
I have certainly seen cases where the other parent has changed their mind.
Pretty sure most of mine were from an early age, as has been said most are have you had them not when.
From what I have seen most waivers seem to get approved but of course it adds time to the process and a chunk of change.
You are being deployed?
There is no set time, you can expect it to be quicker.
Then of course there is the rest of the process, some Consulates are quicker than others.
Call around is all I can suggest, might be variable as above depending on why you speak to.
I know there was a place near Denver that served the Spanish speaking community who would do it, now do not know.