First, I would disconnect the downstairs bell. Next, test the upstairs bell. Does it still work? Next, connect the second bell in-parallel to the first with less than 5 feet of wire and test the two bells. Do they both ring or try to ring? Hopefully. Next, I would try a more powerful transformer or power source. Do you have a small automotive battery charger (2 or more Amp is plenty), or you may be able to use battery(s). Eight 1.5 volt "D" batteries in-series will produce 12 volts. Leave the two bells connected in series upstairs. Disconnect the transformer that feeds your bell(s) and connect the alternate power source. Now do both bells ring properly? If so all you need to do is buy a more powerful 12 volt transformer.
I think your transformer may be partly shorted out. It doesn't create enough of an electrical field for the electromagnetic to slam the hammer into the bell. And when you release the button, only the downstairs hammer moved far enough to swing back and strike the second bell. You either have your wiring crossed somehow, the two bells wired in-series, or the transformer (power source) is bad or not sufficient.
Retail outlets like Radio Sh__ carry 12 volt transformers. You probably want one that supplies more than 1 amp.
I may have been misleading when saying 2 transformers in-phase, because it would be better to increase the size of one power supply for both bells.
For what its worth, in-phase refers to the black or hot AC wire going the same terminal, and the white or neutral wire going to the opposite terminal on each of the transformers. You would have to keep the outputs of the two transformers in-phase as well. It becomes more complex with 2 transformers so try one bigger power supply.