At what point after separating terumot and ma'aserot can I eat the produce?

Question

I understand that after the produce is designated and named, it is no longer tevel and can be eaten. I  don’t understand how this works in my case. And at what point practically, can produce be eaten.

Am I to understand that in the event I pick one cucumber, that I will be required to physically cut (take the tithes on) one cucumber that is only going to be consumed later that night or the next day? Please clarify.