The standard definition of the path integral in Quantum Mechanics usually goes as follows:

Let $[a,b]$ be one interval. Let $(P_n)$ be the sequence of partitions of $[a,b]$ given by $$P_n=\{t_0,\dots,t_n\}$$ with $t_k = t_0 + k\epsilon$ where $\epsilon = (b-a)/n$, and $t_0 = a$, $t_n=b$.

Let $\mathfrak{F}: C_0([a,b];\mathbb{R}^d)\to \mathbb{C}$ be a functional defined on the space of continuous paths on $[a,b]$. One defines its discretization as the set of functions $\mathfrak{F}_n : \mathbb{R}^{(n+1)d}\to \mathbb{C}$ given by $$\mathfrak{F}_n(x_0,\dots,x_n)=\mathfrak{F}[\xi_n(t)]$$ where $\xi_n(t)$ is the curve defined by taking the partition $P_n$, defining $\xi(t_i)=x_i$ and linearly interpolating between the points - in other words $\xi(t)$ is for $t\in [t_i,t_{i+1}]$ the straight line joining $x_i$ and $x_{i+1}$.

One defines the functional integral as the limit $$\int_{C_0([a,b];\mathbb{R}^d)}\mathfrak{F}[x(t)]\mathcal{D}x(t)=\lim_{n\to \infty}\int_{\mathbb{R}^{(n+1)d}} \mathfrak{F}_n(x_0,\dots, x_n) d^dx_0\dots d^dx_n$$

if it exists.

In the case of interest for physics one has $\mathfrak{F}[x(t)]=e^{iS[x(t)]}$ or rather $\mathfrak{F}_E[x(t)]=e^{-S_E[x(t)]}$ the euclidean version.

So by slicing the time axis into equal subintervals, one converts the functional to a sequence of functions, integrates those and takes the limit.

This is the construction outlined for instance in Peskin's book or Sakurai's book, just rewritten in a more "mathematical" form.

Now, if on the very first step we choose another sequence of partitions $(P_n)$ such that the sequence of partition's norms $|P_n|\to 0$ as $n\to \infty$ but which is *not* the sequence of equal subintervals, would the resulting path integral be different?

I don't see reason why it should be equal. The intervals endpoints are distinct, the interpolations are distinct, hence the maps $\mathfrak{F}_n$ are distinct.

If it is I think this is a big problem. After all, the way we are slicing the time axis is arbitrary and just one trick to make the problem easier to deal with.