Is it permissible for my paternal uncle to take things from my father's estate after his death, based on an alleged will for which he has no proof, knowing that some of my father's possessions were in my uncle's name?

What your uncle claims is a baseless assertion unless you corroborate it or he proves it with legitimate evidence, as the Prophet, peace and blessings be upon him, said: "Were people to be given according to their claims, men would claim the blood and wealth of others. But the burden of proof is on the claimant, and the oath is on the one who denies."

Therefore, your uncle is not entitled to enforce his claim unless he provides proof or you yourselves confirm it, provided you are adults of sound mind. Otherwise, it does not apply to the share of one who is not an adult of sound mind.

Given that Islamic law encourages reconciliation and upholding kinship ties, it is preferable for you to reconcile with him and settle the matter amicably. Otherwise, you may refer the matter to the Sharia court.