Yes you can usually still move your fingers when you have a broken wrist.
Sometimes you can move both your fingers and even still move your wrist some when you have a broken wrist.
If you have a broken wrist you should see a doctor about it to make sure it does not become infected or cause worse problems.
You should avoid using your wrist for awhile until the broken wrist heals to avoid further injury.
When I had my broken wrist it took 8 weeks to fully heal but it started healing within a few weeks of getting it fixed by the hospital.
The doctor advised not using that wrist for at least a few weeks and within 4 weeks the broken wrist was nearly healed.
I was able to still move my fingers some but I avoided using that wrist and even the fingers for at least 4 weeks to allow the wrist to heal.
If you notice any swelling or more pain in the wrist go to the emergency room or urgent care or see your doctor because it could become infected.