Grammar – ‘Once I Finished’ or ‘Once I Have Finished’?

Both are correct.

have finished is present perfect of the verb finish. Present perfect is used when an action completed in the past has an effect that lasts to the present. This version focuses on something that happened in the past, and its effect on the current situation.

In am finished, finished is an adjective: am describes the situation now, without any regard to what happened in the past.

This NGram shows that there is a growing difference between British and American usage.

have finished is the older and possibly more formal form, with am finished only starting to appear in about 1900. In the UK, the new form had a limited life and is now in decline: in the US, however, it has grown steadily in popularity and has now overtaken have finished.

I am British English and could easily imagine my grandmother saying am finished, but I would prefer to use have finished.

Effectively the difference is that only one is grammatically correct.

I'll do it after I have finished that

This is grammatically correct, it correct refers to future tense. I will do this thing once I HAVE completed the other thing.

I'll do it after I finished that

Grammatically this isn't correct, what could be used instead is

I'll do it after I finish that

This way there is no need for "have" but both mean similar things, with have suggests you're not working on the focus right now but will be soon in the future, using just finish suggests you're currently actioning this.