Some Questions and Concerns you Should think about....
Did you Put the Battery the Right way?
(1) Sometimes You might mistakly put the battery the wrong way. Which might make it not function properly, or even function at all. So when you put your new battery in make sure its Face Side up. Then your tamagotchi should be working properly. But according to your problem I don't think its that.
Did you put the Right Battery?
(2) As Binary mentioned you need a CR2032 battery for your Tamagotchi. As Binary also mentioned you may find similar battery look-a-likes,that might not be suitable for your Tamagotchi. So always remember what kind of battery you need. What I always do is I bring in my old tama battery and give it to the person, so they can get exactly that same battery.
Did you buy a Cheap Battery?
(3) Sometimes, you can buy [ even myself ] buy cheap batterys. Cheap batterys can cause the battery to die way faster, then a durable non-cheap battery would. Even if the package mentions that it is ''A Long-Lasting'' Battery, it might not be. You can find cheap batterys at the Dollar Store. Though some dollar store batterys work properly, but some may not. Most likely a Dollar Store battery won't last you very long. Some batterys may only go up to 2 weeks. When I buy [sometimes] Dollar store batterys, it can last up to 2 months. But sometimes you can find cheap batterys at big stores as well like Wal-Mart. Just because it isn't a ''Dollar Based'' Store, some batterys, especailly no-name brands may cause battery failure.
Is the Battery Dirty/Dusty?
(4) Sometimes, you Just have to take out the battery and blow it slightly against the battery and on the back of the tamagotchi. Then when you put back in the battery just simply Download.
I hope I helped. It could also mean your Tamagotchi is Broken. Try these steps and tips. Also use the tips the others have mentioned and see if you get results. If not your tamagotchi is unfortunatly broken!
Hope I helped,
~tama_princess~