Learn Spanish (Latin America): ¡Ojalá tuviera mi licencia hoy!

Even though the wish is about today, using ojalá with the imperfect subjunctive expresses an unfulfillable or impossible wish in the present. It translates to 'I wish I had' (notice we use the past tense 'had' in English too for impossible present wishes!).

What would it mean if I used the present subjunctive and said ¡Ojalá tenga mi licencia hoy! instead?

Using the present subjunctive (tenga) changes the sentence from an impossible wish to a real possibility. It would mean 'I hope I get my license today!' or 'Hopefully I have my license today!', implying there is an actual chance it might happen.

Could I use tuviese instead of tuviera here?

Yes, tuviese is the alternative form of the imperfect subjunctive and means the exact same thing. However, the -ra ending (tuviera) is much more common in everyday Latin American Spanish.

Why isn't there a que after ojalá here?

Unlike conjugated verbs of hope (like espero que), the word ojalá does not strictly require que before the subjunctive. You can say ojalá que tuviera, but dropping the que is completely correct and very common.