The fact that $\bf N$ is well-ordered and totally-ordered poses no obstruction to putting a uniform probability distribution on the naturals. Rather, it is the countableness of $\bf N$ that precludes such.
The reason is that by definition such a probability measure $\mu(\cdot)$ must be countably additive:
$$\mu\left(\sum_{i=1}^\infty A_i\right)=\sum_{i=1}^\infty \mu(A_i)$$
for any family $\{A_i\}$ of pairwise disjoint subsets $A_i\subseteq\bf N$. Let $p\in[0,1]$ be the probability of picking a given individual natural number; then
$$1=\mu({\bf N})=\sum_{n\in{\bf N}}\mu(\{n\})=\sum_{n=1}^\infty p=\begin{cases}\infty & p>0 \\ 0 & p=0,\end{cases}$$ which is absurd.
At this point there are a number of possible ways to go around this obstacle. The typical way is to settle for a finitely-additive but not countably-additive measure, namely asymptotic density. This can be defined as $\mu(A):=\lim_{n\to\infty}|A\cap\{1,2,\cdots,n\}|/n$. The natural density exists only for particularly nice sets. It might be more natural to augment this device for $\bf Z$ instead of $\bf N$. We see from this that the ordering is actually helpful rather than harmful, and it predicts that the events of prime divisibility are independent. That is, $\mu(p{\bf Z}\cap q{\bf Z})=\mu(p{\bf Z})\cdot\mu(q{\bf Z})$ for distinct primes $p,q$.
Another route around the obstacle is to give up uniformness instead of countable-additivity - here we are channeling the combinatorial insight of Rota IIRC. Associating a probability distribution to the naturals is the same as associating positive values $a_n$ to each $n\in{\bf N}$ such that $\sum a_n=1$. In this way we obtain a parametrized family of probability distributions called zeta distributions:
$$\mu_s(A)=\frac{1}{\zeta(s)}\sum_{a\in A}\frac{1}{a^s}.$$
In this case, we again have divisibility by distinct primes as independent events. Often we can take the limit as $s\to1^+$; if the natural density of $A$ exists then $\lim_{s\to1^+}\mu_s(A)$ exists and is equal to it, but the limiting zeta measure is actually more general in that it exists in some cases where the natural density does not, so it is an even better choice. A related concept is Dirichlet density.
A third route around the obstacle is to change the setting. As remarked in the comments, the finite quotients of $\bf Z$ have uniform probability distributions and divisibility statements make sense in these contexts, but each quotient captures only so much information about $\bf Z$ locally - what if we could "patch" together the information from each local piece in a way that we obtain something even bigger than $\bf Z$ which admits a uniform probability measure?
We can, in fact, do this, with the profinite integers $\widehat{\bf Z}=\varprojlim{\bf Z}/n{\bf Z}$. As a profinite group under addition, it is topologically compact and carries a unique normalized Haar measure, this gives rise to a uniform probability distribution on $\widehat{\bf Z}$. In this setting $\bf Z$ by itself has measure $0$, but the fact that it's dense in the profinite integers suggests we should instead define a $\mu$ on $\bf Z$ by invoking topological closures: $\mu_{\bf Z}(A)=\mu_{\rm Haar}(\overline{A})$. Here once again the events of divisibility by distinct primes are independent. I'm curious myself of what other properties it has.
This is also a group-theoretic way of informally investigating, or generalizing, the question of what probability two "random" integers have of being coprime. Integers $n,m$ being coprime is equivalent to $\langle n,m\rangle=\bf Z$ i.e. $n$ and $m$ generate $\bf Z$ under addition, by Bezout. The probability $k$ randomly chosen elements of $\widehat{\bf Z}$ topologically generate the whole thing is $1/\zeta(k)$.
Even more tangentially, we say a profinite group $G$ is positively finitely generated (PFG) if for some $k$ there is a nonzero probability that $k$ randomly chosen elements topologically generate the whole group. This moves into the territory of zeta functions of groups. As it turns out, a group is PFG iff it is PMSG - the number of maximal subgroups grows at most linearly with index. There is a conjecture (Lucchini) that the zeta function of a f.g. profinite group is rational iff its Frattini subgroup has finite index.
