If you won both cases, you probably have a lawyer. So it would be best to ask him or her. Your attorney can tell you how long it might take the court to process an order and send it to the driver's license officials.
You should ask your attorney if you have something like we do in California: a factual finding of innocence. That means: a judge, upon proper motion, can declare that you are factually innocent (a granting of the motion means that there would not even be an arrest record), in California. It is a difficult burden to meet, but you can ask your attorney, nonetheless, if such a provision exists in your state.
Once the implied consent victory is entered into the court's system, it will electronically transmit to the DMV. It should only take a few days for them to process their files and remove the event from your record.
You can also get your license reinstatement fee refunded, but it is a pain to deal with.
If you don't want to bother your lawyer, you can call DVS directly at (651) 296-2025. They can let you know where they are in the process.
Down the road, you may want to order a copy of your drivers record, so you can check for yourself to make sure the incident has been removed from your record. The paperwork you will need is available here:
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