A lot of this will have to be speculation since most of these things are never explicitly explained, but we do have a couple of clues.
During the episode
SHINJI: What is this? What did you do, Father?!
HYUGA (OFF): Reception of the signal, confirmed.
IBUKI (OFF): Control system switch is complete.
MAN (OFF): All nerves linked directly to the dummy system.
32.8% of Emotional Elements are unclear. They can't be monitored.
IKARI: Irrelevant. Release the system. Commence attack.
2 things are probably important here. One is that Shinji is in the plug and the second is that there was some technobabble about some "emotional elements". We know Yui is within Eva-01 and her emotional attachment to her son is partly why the Eva+Pilot pairing ends up being so powerful. That's probably part of what's going on here, keeping in mind that during this dialog Eva-01 is actively in a wrestling match with Eva-03, an Angel, and in grave danger. It's entirely possible that Yui "let" this happen. There's some other possible actions, of course, like the Eva going berserk like it has in the past to save Shinji. Maybe this was just the path of least resistance.
However, during episode 19:
Commence A10 nerve connection.
So it's not going to work anymore.
Pulse flow is reversing!
Unit 01 is rejecting the neural connections!
No, that can't be possible!
Yes, it's rejecting me.
Abort activation. Sortie Rei in Unit 00.
Re-activate Unit 01 with the dummy plug.
and after they swap her out and insert the dummy plug:
The pulses have vanished. It's rejecting the dummy plug.
It's no use! Eva Unit 01 won't activate!
The dummy, Rei...
It won't accept them?
So we know Rei has synced with Eva-01 before, but for some reason the Eva itself is refusing Rei. Rei, being the dummy system's core, was also refused (as Ritsuko notes to herself). Gendo notes that the Eva is specifically rejecting him, which seems to be plausible since both Rei and the dummy system has successfully syned in the past but for some reason, this particular moment, Gendo and Yui's agenda diverge.
We can only speculate what those plans were, and what made them split at this particular point. Given the flashbacks and the retro episode, it's probably the case that their big plans involved the 3 of them: Gendo, Yui, Fuyutsuki, and it probably involved Shinji to some significant degree. But it seems to me that Yui wanted Shinji specifically, and that part of her plan was to set this in motion.
Later in Episode 19:
SHINJI: Move! Move! Move! Move! Move! Move! Move! Move! Move! Move!
Move! Come on, move! If you don't move now, if you don't do it now,
everyone's going to die! I don't want any more of that! So... PLEASE
IBUKI: Eva has re-activated!
IBUKI: No way, this is unbelievable. Unit 01's synch rate is over
RITSUKO: Does that mean she's really awakened?
Having Shinji as the pilot, who seemed to have been able to activate Eva-01 instantly without the aid of plugsuits or anything, they quickly make some short work out of the most powerful Angel to date, runs out of power, and Yui (possibly with the help of Shinji) turns the berserker thing up a notch and syncs with him at 400% and "awakens". Not entirely sure what it means to be in an awakened state, but I think it's probably safe to assume that it involved acquiring an S2 organ from Zeruel (by eating it) and that all this wasn't going to happen with either Rei or the dummy plug in the pilot's seat.
(Speculation) So Yui's gamble that Shinji would "do the right thing" and come back to pilot paid off as now Eva-01 has unlimited energy with an S2 organ. This step in their "big plans" seemed to have split between Yui and Gendo, although in episode 20, Gendo recognizes what has gone on and does't seem to disapprove:
Episode 20, watching the action
FUYUTSUKI: It's begun, hasn't it?
IKARI: Yes, it all begins here.
And of course, Seele are furious over this, it's clearly not part of their plans
COMMITTEEMAN ?: The Eva series aren't capable of generating S2 engines
COMMITTEEMAN ?: We never imagined it would take one into itself this
COMMITTEEMAN ?: This incident is at extreme odds with our script.
COMMITTEEMAN ?: It will not be easy to correct for this.
COMMITTEEMAN ?: Were we not mistaken to have entrusted Nerv to Gendo
Ikari in the first place?