Am I obligated to offer an expiatory oath (kaffarat yamin) or give charity with the amount I stipulated to donate if the doctor informs me that my finger will not be amputated, knowing that my recovery is not yet complete and I might need the money for treatment?

If you made your oath conditional on the doctor's informing you that your finger would not be amputated, and you intended complete recovery in the future, then nothing is incumbent upon you if the doctor did not definitively state that, because your oath or vow's condition was not met. However, if you made your oath conditional merely on the doctor's informing you at that time that it would not be amputated, regardless of the future, then you are obligated to give the amount in charity. This vow of yours does not lapse with delay, and if you do not give charity despite being able, then an expiation for breaking an oath becomes incumbent upon you due to the delay. But if you do not give charity due to incapacity, then no expiation is incumbent upon you, though it is preferable to give charity whenever it becomes easy.