Uniform mortality problem for Turing Machines
The mortality problem is undecidable (P.K. Hooper, Th eUndecidability of the Turing Machine Immortality Problem (1966))

The uniform mortality problem undecidability follows from the following:

Theorem: A Turing machine is mortal if and only if it is uniformly mortal

I found the proof in: Gerd G. Hillebrand, Paris C. Kanellakis, Harry G. Mairson, Moshe Y. Vardi: Undecidable Boundedness Problems for Datalog Programs. J. Log. Program. 25(2): 163-190 (1995)

Proof: Let $\delta_1,...,\delta_n$ be the possible transitions of $M$. We call a sequence $\delta_{i_1},..,\delta_{i_k}$ of transitions consistent if it reflects a computation of $M$, i.e., if there exists a configuration $c = (l,s,r,q)$ ($l$ left tape, $s$ symbol under head, $r$ right tape, $q$ current state) from which $M$ will execute that sequence.

Now arrange all consistent transition sequences in a (possibly infinite) tree, with the empty sequence at the root and each node extending the sequence at its parent by one transition. This tree is of bounded degree. Also (1) $M$ is mortal iff there is non infinite path in the tree, and (2) $M$ is uniformly mortal iff there are no arbitrarily long paths in the tree. By Konig's Lemma (in a tree of bounded degree there is an infinite path if and only if there is a family of paths of unbounded length), these two conditions are equivalent.