I am not 100% that Blizzard III changes to UI faster, but I am 100% sure that Fire III changes to AF3 before the cast even finishes.
Easily testable and I even spotted with yoshi's titan video. I paused the youtube and got him midcast with FIRE III and no buff even though his UI3 buff had 8 seconds remaining when he started the cast.
This is the case on live, and you can confirm it yourself. You will lose UI3 during the cast of Fire III and get AF3 afterwards.
This means you can't get the double reduced cast time and you can't get mana back after casting flare, hence no more double flare.
The consequence of this is that our UI3 phase is now shorter. Before our UI3 phase lasted even after the Fire III cast allowing us to be fully topped off.
Now it ends before the Fire III even finishes, which is a significant reduction.
Before, when you were unlucky with ticks, you'd still at least get a tick after Fire III.
Now that's not possible, so if you're unlucky with ticks you will not be at full mana.
However, it's clear they never really meant for us to get anything from UI3 after casting Fire III.
The mana ticks were not synced and that is false information lead by someone who casually watched the live letter and said that.
Then it got spread even though those of us who watched the youtube video and analysed it carefully conclusively disproved it.
Now the patch is live and anyone can test it, but people are still talking about syncing when the ticks are, in fact, as fast as they always were.
Our UI phase is just shorter because the transition is quicker.
I am 100% sure your own test could not have conclusively proven that UI3 ticks are longer.
I am also sure you did not really look at your UI/AF buff when casting fire 3 to transition to AF3.
EDIT:
The fact that you can get ticks almost instantly after switching to UI3 disproves any notion that the mana ticks have changed (where people mostly say it takes 3 seconds for every tick).
Also, the ticks have always been on server time.
Go back to any of the Black Mage threads and this has been said repeatedly. It always ticks every Eorzea minute.
People have been trying to figure out if they could use that to their advantage since launch.
It ticks likes that before 2.1 and it ticks like that after.
This is how it's conclusively proven that mana ticks were unchanged because we already have a timer for them.
Coupled with a change to the AF transition that we haven't fully calculated yet, there's no reason to believe it has anything to do with mana ticks.