Sometimes the ball gets stiff - I've had problems with various water bottles. If the water isn't coming out you need to change it asap and meanwhile put a little saucer or shallow dish of water out so your hammy doesn't go without water. It needs to be very shallow so they're not likely to fall in it and get wet or worse. But it can still hold a reasonable amount widthways so they can lick at it from the edge.
I settled on these bottles and used to keep two in the cage all the time in case one stopped working.
If you get two now you only pay one set of postage plus you can have a spare or have two in the cage. I've never had any problems with them letting out water plus they are easy to use as have a clip that attaches inside the cage and the bottle just clips into it, so easy to take in and out. I get them from Porton because there aren't many places stock them and the price is cheaper than Amazon. They usually deliver within 1 or 2 days. The 75ml mouse sized one is a good size for a hamster.
Fpi 4660 Drinky75 Drinky B.
When filling the bottle, (current one as well) don't fill it right to the top. Fill it about 3/4 full. An air gap at the top seems to help the water flow better.
With your current bottle you could try this now and also try unfastening the screw top a little bit at a time (ie looser) until water starts to come out. Too far and the bottle will just drip and all the water run out quite quickly. But it might get it working temporarily. If the metal ball in the spout is stuck or gone stiff that might not make a lot of difference. I would definitely replace it asap though - they can die of dehydration.