Fig.: when the situation gets critical and a decision or an action must be made

When push comes to shove, my boss is second to none in finding and adopting the best plan of action

Quando arriva il momento di agire, il mio capo non è secondo a nessuno nel trovare e nell'adottare il miglior piano di azione

There is a lot of talk about gender equality, but unfortunately it still happens that, when push comes to shove, a man is still preferred over a woman for a promotion

Si parla tanto di parità di genere, ma purtroppo succede ancora che, arrivati al dunque, l'uomo venga preferito a una donna per una promozione

 I was going through difficult times and only a couple of “friends” showed up to help me. It's really true that you recognize true friends only when push comes to shove

Per me erano tempi difficili e solo un paio di “amici” mi hanno offerto il loro aiuto. È proprio vero che solo alla resa dei conti si riconoscono i veri amici

The origin of this expression is unclear. The most accepted theory is that it originated in the late 1800s from African-American English.
A simple, literal explanation could be that Pushing and shoving is a way of referring to man-to-man fighting, so when a push becomes a shove (an exaggerated - bigger and harder - push), it means that a fight is about to break out.