First of all, heartiest congratulations to you and your partner. Secondly, yes it is correct that one should get one’s marriage registered where they are located. But it is not the complete thing. There are two aspects: marriages solemnized under Hindu law can be registered by the sub-registrar under whose jurisdiction the marriage has been solemnized, or where either of the bride or groom resides for a period of more than 6 months. So if you are getting married under Hindu law, you can get your marriage registered where your marriage takes place, or in Mumbai but not in Pune since your wife has resided there only for a period of 5 months which is less than the required period of 6 months.
But if you are getting married under the Special Marriage Act, you can get your marriage registered either in Mumbai or Pune as under this Act, marriage can be registered where either the bride or groom resides but not where the marriage takes place, since under this Act, marriage is solemnized only after it’s registration.
Nowadays, marriages can be registered online as well. The couple can fill the form at the website of the state in which their marriage has taken place. Then on a specified date, the Registrar of the area would call the couple to verify their documents, alongwith witnesses and register the marriage.
Hope this satisfies your query.