The nuance difference is very subtle but both are natural. "My visa got approved" is the most common way to express that is was approved by the relevant authority. "I got my visa approved" kind of sounds like you had to push for it to get approved by contacting the relevant authority being of a problem or delay. So definitely using "My visa got approved" would be best.

I won't say "My visa got approved" is the same as "I got my visa approved" because "My visa got approved" is more of a general statement without the "subject" no indication of who got your Visa approved. But "I got my visa approved" indicates who got the Visa approved.

Yes, this would be a correct way to say this. However, a better way would be "my visa was approved" or "my visa is approved". Hope that helps.

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