We should first discuss whether you had a right to break the chain and move the bicycle; that question has some impact on your question, as well. In general, one may “take the law into his own hands” to rectify/prevent damages in a manner that causes the least necessary loss to the (potential) damager.[1] If necessary, one may break things that prevent him from passing through a domain in which he deserves access.[2] It appears, then, that the ongoing inconvenience and potential danger the bicycle caused justified your taking action. If you exhausted reasonable efforts to find the owner and have him remove the bicycle, it was proper to break the chain and relocate the bicycle.
At the point that you have already broken the chain and moved the bicycle, the question is whether you have the normal responsibilities of hashavat aveida,[3] which includes taking actions to preserve the object until the owner is found.[4] Perhaps you could claim to be exempt because there is no lost item, as the owner presumably knows where the bicycle is. Indeed, when someone purposely leaves his object somewhere, no one needs to move it, nor should he, with the intention of returning it.[5] However, this is not relevant in this case because when the owner left it, the bicycle was chained up, and now it is not. In this new situation, it is likely a service to the owner to take steps to protect the bicycle. This resembles the case of one who hid an object in a garbage dump that was not slated to be emptied. If, subsequently, the dump is going to be emptied, the object is no longer safely hidden, and there is an obligation of hashavat aveida when feasible.[6]
However, a different exemption from hashavat aveida seems to apply in this case. The gemara mentions in a few places[7] the notion of aveida mida’at (literally, an intentional loss). The common denominator of the cases is that an owner purposely left his object in a situation in which its chances of remaining intact and in his possession are poor. Consequently, there is no requirement of hashavat aveida. The Rambam[8] derives this ruling from a pasuk – hashavat aveida applies when the object was “lost from him,”[9] as opposed to an intentional loss.
The Rambam[10] says that although there is no mitzva of hashavat aveida in the event of aveida mida’at, the finder may not claim the object for himself. The Tur[11] says that the finder may even take the object because we consider the owner to have been mafkir (relinquished his rights to) it. According to the Rambam, there appears to be a rule that one does not have to go to greater efforts to protect someone else’s possession than the owner himself did.[12] Similarly, in your case, in which someone left his bicycle in such a manner that he should have expected people to eventually exercise their right to take action, he knowingly exposed it to the prospect of being unprotected. Thus, you are not required to take steps to remove the danger that he should have anticipated. The Tur probably agrees to this concept in those cases of aveida mida’at in which hefker does not apply.[13]
In summary, you are probably not required to obtain a chain for the bicycle you moved, especially if the owner still has good chances of retrieving it at its present location. On the other hand, you do not know what circumstances caused the owner to leave the bicycle as he did for as long as he did. Therefore, it would be a responsible step and a nice gesture to protect it.