Getting Pushed while in the middle of Two Weapon Fighting

Yeah, Cody is right. The push gets applied immediately, and then you go back to your normal actions (which in this case is your offhand attack).

Not sure exactly what you are asking here, but. . .

A Free Strike is a Combat Maneuver,
and as such it may not be performed during the Recovery
period of another Combat Maneuver. The character
makes a Basic Attack (melee only) against the target, but
may not perform a Trivial Action as part of that attack.
The attacking character Pushes his Clock by the Basic
Attack’s Speed instead of advancing it as normal.

A character can only be subject
to one Push effect at a time, though the longest such
period will always apply. For example, if a character is
subject to a Push 3 effect, then a Push 5 effect 2 Ticks
later, that character only suffers an additional 3 Ticks
of Push (the Push 3 effect is superseded by the Push 5
effect, effectively increasing the duration of the original
push by the difference.)

As such, you cannot willingly be subjected to more than one push at a time. If you are subjected to a second push, you are Pushed by whatever push effect is higher. For example, if you have one more Tick of being Pushed, and are then suffer Push 4, you now have Push 4.

I think that is a fair interpretation. According to the rules as written, the player would continue their off-hand attack as soon at their Push is done, but as a GM I would say that your interpretation is more realistic. Basically, to take that free-strike they are abandoning their maneuver midway through according to my view.

Free strike is a Combat Manoeuvre with a recovery of 0. See Errata/FAQ.

It can always be taken unless under a Push.

Also, until the second TWF strike is taken, there is no recovery (as Speed/recovery is calculated after the second attack).

Errata FAQ; you can still make a Combat Man with a recovery of 0 whilst under Recovery.

In this case, there is also no recovery to assess, until the second action is taken.