Time Complexity of Universal Turing Machine Simulations and the Time Hierarchy Theorem
I'm referring here to the proof of the hierarchy theorem as I am familiar with it, in which I don't see the problem you mention.

We define the language
$L=\{(M,w): M $ does not accept $(M,w)$ within $\le \frac{t(n)}{|M|^3+|M|\log t(n)}, n=|(M,w)|\}$ (where $t$ is the time function we are working on).

We show that $L$ is decidable in $O(t(n))$ using the universal machine, and in the universal machine each step is indeed dependent on the size of the machine, which is why we have $|M|^3$ in the denominator (the 3 is just some upper bound on the computations needed to simulate a step).

For completeness of the answer: when we try to reach a contradiction, then we assume that there exists a machine $T$ that decides $L$. After this assumption, the encoding of $T$ is fixed, and then a simulation of a single step really does take constant time, and we can reach a contradiction by taking a long enough $w$ to increase the length of the input without changing $T$.