All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
First of all, we would like to draw your attention to the fact that you should not be attached to this cat to this extent in a way that it preoccupies you and saddens you. This is a dispraised matter as it prevents you from what benefits you in your worldly and religious matters.
As regards protecting the animal, taking care of it and repelling harm from it, then this is something religiously required. Indeed, Allaah was thankful to a prostitute and forgave her because she gave water to a thirsty dog as in the Hadeeth reported by Al-Bukhari and Muslim . On the contrary, a woman entered Hellfire because of a cat; she tied it until it died, she did not feed it and she did not let it eat from the vermin of the earth as in the Hadeeth reported by Muslim . For more benefit, please refer to Fatwa 83830.
Being concerned that this cat would not fall into the hands of abusive people is something good. So, if you predominantly think that the cat shelter to which you gave the cat will take care of it, then you are acquitted from sin. You may ask them to keep it with them until you recover from your disease and then you would take it from them in order to keep it with you or give it to whoever may take care of it.
With regard to your question about what will happen to the animals in the Hereafter, then the answer is that Al-Haakim reported in “Al-Mustadrak” that Abu Hurayrah said: “There is no animal or bird but will be resurrected on the day of Judgment, then Allaah will command them to take revenge (from the one among them which oppressed them) to such an extant that a hornless sheep will be treated just and given its revenge from the horned sheep (which did injustice to it in its life) Then Allaah would say to them: “Turn into dust.” At this point, the disbeliever will say: “Woe to me! I wish I were dust." Then he (Abu Hurayrah ) said: “If you wish you may recite the saying of Allaah (which means): {And there is no creature on [within] the earth or bird that flies with its wings except [that they are] communities like you. We have not neglected in the Register a thing. Then unto their Lord they will be gathered.}[Quran 6:38]”
Such a statement has the ruling of being Marfoo’ (i.e. attributed to the Prophet ), because such matters cannot be spoken of on the basis of personal opinion.
This is evidence that the animals will not be in Paradise on the Day of Judgment. Allaah is over everything Omnipotent, but He decreed this for the animals. Therefore, your supplication to Allaah to make your cat with you in Paradise is not an appropriate supplication.
Allaah Knows best.