Which of the following is correct?
- I will update you once I reached my home.
- I will update you once I reach my home.
- I will update you after I reaching my home.
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Among the three samples you have provided, the middle one is correct:
I will update you once I reach my home.
The first one is wrong because you are mentioning there that you'll do something in the future after you do something in past(which of course doesn't make sense).
Same situation arises in the third one.
Again, as @StoneyB mentions:
get is a much better choice here.
The last one is wrong, since you don't say "I reaching my home"; it would be "I am reaching my home." but you don't say "I will update you after I am reaching my home."
The correct one is "I will update you once I reach my home." I would rather say "once I arrive home," but I think that is a matter of style.
According to the Cambridge Dictionary, it is actually possible to do things in the future after having done some in the past (what?!?):
We use once as a conjunction meaning ‘as soon as’ or ‘after’:
Once I’ve picked Megan up, I’ll call you.