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.