How can $\mathbb Q$ be countable, when there is no "next" rational number?

The 'next' rational after $\frac{3}{1}=3$, in this context, is $\frac{3}{2}$. If you like let $\varphi$ be the bijection $\varphi:\mathbb{N}\rightarrow \mathbb{Q}$. Then we do have

$$\varphi(12)=\frac{3}{1}=3$$ and perhaps we might write $$S^\varphi_{\mathbb{Q}}(3):=\varphi(S(\varphi^{-1}(3)))=\varphi(S(12))=\varphi(13)=\frac{3}{2},$$ where $S:\mathbb{N}\rightarrow\mathbb{N}$ is the successor function $S(n)=n+1$ and we might call $S^\varphi_\mathbb{Q}$ is the successor function of $\mathbb{Q}$ with respect to $\varphi$ defined by

$$S_\mathbb{Q}^\varphi:\mathbb{Q}\rightarrow \mathbb{Q},\,\,q\mapsto \varphi(S(\varphi^{-1}(q)))$$

First let's ask a different question.

How can $\Bbb Z$ be countable, if every number is a successor in the integers, whereas in $\Bbb N$ there is a number which is not a successor?

The answer to this question, and the answer to the question why is $\Bbb Q$ countable are the same answer: Because a bijection is not order preserving.

Cardinality of a set is what remains when you "shake off" any structure it has or might have had. Namely, cardinality is determined by functions which are not necessarily preserving any structure whatsoever (order, addition, multiplication, etc.).

It is true that it's harder to see why $\Bbb Q$ is countable compared to $\Bbb Z$, especially when $\Bbb R$ is not countable and both are dense ordered sets. But if you understand why $\Bbb Z$ is countable, then you shouldn't have any difficulties understand the case for $\Bbb Q$: we can prove that there is a bijection between $\Bbb Q$ and $\Bbb N$, and therefore it is countable.

When you say "next", you are probably transferring a feature from $\Bbb N$ that $\Bbb Q$ lacks, namely well-orderedness in the natural order.

But $\Bbb Q$ can be given other orders that allow each element to have a successor, and your partial organized list is a case in point. Any bijection with the naturals would induce a well ordering on $\Bbb Q$ by simply enforcing the order on the natural numbers on their images. Conversely, you can create an order on the naturals by finding a bijection of $\Bbb Q$ with $\Bbb N$ and then using $\Bbb Q$'s natural order to put a new order on the natural numbers in which elements wouldn't have successors.

Having successors for elements in a particular order simply doesn't have much to do with cardinality. You can have sets of large size where elements have successors, and sets of large size that don't.

Density and cardinality are two distinct types of "largeness." Lack of successors does not cause the cardinality to go out of control. It just speaks to the organization of the ordered set.