I planted a few different varieties this year and I didn't see flowers on at least one variety. About a week ago it seemed like the flowerless plants were just starting to die off (plus I was eager for potatoes) so I dug them up and found a ton of medium to very small, very tasty potatoes. I'm sure if I had left the plants another few weeks or more I would have found a ton of medium to full sized potatoes. For reference I started these potatoes on April 12th, so about 13 weeks of growth time.
I've read you can harvest potatoes after about 10 weeks for immediate consumption, called "new" potatoes. They are smaller with thin skins and amazing flavor. From my experience I would recommend a few extra weeks because some of my potatoes were quite tiny.
Alternatively, you can let the potatoes keep growing until the plants die back or harvest right before the first frost. Then they need to cure for about two weeks before you put them away for winter storage.
Some people like to do a combo and collect a few potatoes to eat right away and then leave the rest for later. If you gently dig in the soil at the base of the plant you will find the oldest and largest potatoes. Take a few and replace the soil and leave the rest for a future date.