According to the beautiful Noether’s theorem, a conservation law is associated to a specific symmetry [...] Is there a symmetry associated with the conservation of those quantum numbers?
Yes, there are. Judging by your question I think a very brief and basic comment on how symmetries work in QM will be helpful.
Let $\mathcal H$ be the space of states of your system, $G$ the symmetry group of your system. Elements $g$ of the group $G$ act on the states $|\psi> \in \mathcal H$ through
$$|\psi> \rightarrow U(g) |\psi>$$
and on observables through
$$
A \rightarrow U(g)\,A\,U(g)^{\dagger}
$$
Where $U(g)$ are unitary operators that form a representation of G. Usually $G$ is either a Lie Group or a discrete group. If it's a Lie group, under some hypoteses one can write $U(g)= e^{c_a T_a}$, where $c_a T_a$ is a linear combination of (representatives of) the generators of the group's algebra. The number of generators is equal to the dimension of the group, but for the present discussion let's just consider a single generator and let's write just $e^{c T}$. The $T$'s can be taken to be self-adjoint.
Requiring that the Hamiltonian $H$ is invariant under the action of the group means
$$ H = U(g)\, H \,U(g)^{\dagger} = e^{c T} H e^{-cT} \qquad \forall g(\iff \forall c)$$
Taking the derivative of this line with respect to $c$ you get:
$$0= TH -HT = [T,H] \quad \Rightarrow \quad \dot T=-i[T,H]=0 \qquad \qquad (1)$$
And this means that the generator $T$ of the symmetry is conserved (I'm of course talking in Heisenberg picture). Notice that from $(1)$ also the reverse is true: if you have a conserved observable $A$, you can generate a symmetry acting with $e^{cA}$.
This is the link between symmetries and conservation in QM.
There are many details I'm glossing over, and complications may arise, but this should give you the idea.
...charge conservation, lepton number conservation, baryon number conservation TCP...
Charge, lepton and baryon conservation arise from this exact mechanism. The symmetries are the global gauge symmetries of the Lagrangian, and groups involved for example are $U(1)$ for the electric charge in QED, $SU(2)$ for the leptonic number in electroweak theory; their generators (and combinations thereof) give the operators associated with those quantum numbers.
I also noticed that the symmetries associated to the (classical) conservation laws by Noether are the same associations as in the uncertainty principle. Can the two be related in some way, or deduce one from the other?
I think the link you mention is accidental. The uncertainty principle (which is actually a theorem) holds for every couple of observables and states that for every two observables $A,B$
$$ \sigma_A \sigma_B \geq \langle \frac{[A,B]}{2i} \rangle $$
Where $\sigma$ is the square root of the variance, and all means are computed on the same state.
