The solvable Baumslag-Solitar group $BS(1, n) = \langle a, b \, \vert \, aba^{-1} = b^n \rangle$ with $n \in \mathbb{Z} \setminus \{0\}$, has only one $T$-system of generating pairs and any number of Nielsen classes, including infinity, can be achieved for a suitable choice of $n$. This [preprint][7] presents of proof of my claim, see Corollary E. 

The proof is elementary and can be summarized as follows. First identify $G = BS(1, n)$ with $\mathbb{Z}[\frac{1}{n}] \rtimes_n \mathbb{Z}$ where the canonical generator of $\mathbb{Z}$ acts as the multiplication by $n$ on $R = \mathbb{Z}[\frac{1}{n}]$. Then the commutator of a generating pair of $G$ is of the form $(1 - n)u$ for some unit $u$ of $R$ and every unit arises in this way. By Higman's lemma, if two generating pairs $\mathbf{g}_1$ and $\mathbf{g}_2$ of $G$ are Nielsen equivalent then the corresponding units $u_1$ and $u_2$ are related by an element of the form $(-1)^kn^l$ with $k, l \in \mathbb{Z}$. It is not difficult to show that every generating pair of $G$ can be Nielsen reduced to $(a, b')$ for some $b'$ that identifies with a unit of $R$. Now it is immediate to see that if $u_1$ and $u_2$ are related by an element of $(-1)^{\mathbb{Z}}n^{\mathbb{Z}}$ then $\mathbf{g}_1$ and $\mathbf{g}_2$ are Nielsen equivalent. So $G$ has as many Nielsen classes of generating pairs as elements in $R^{\times}/(-1)^{\mathbb{Z}}n^{\mathbb{Z}}$. A careful inspection of $R^{\times}$ shows that the cardinality of the latter is infinite if $n =  6$ and equal to $d \in \mathbb{N} \setminus \{0\}$ if $n = 2^d$. As the group $G$ has only one $T$-system of generating pairs (see e.g, [4]), it answers OP's question with residually finite instances.

There are some classical examples which can provide insight on OP's question. The fundamental group $\pi_1(\Sigma_g)$ of a closed surface of genus $g$ is long known to have a single Nielsen equivalence class of generating $2g$-tuples, see [2, 6].
The family of one-relator groups $T(m, n) = \langle a, b \, \vert \, a^m = b^n \rangle$ for $m,n \in \mathbb{Z}$ has been also extensively studied. It follows from [1] and [5] that each $T$-system of generating pairs of $T(m, n)$ contains a unique Nielsen equivalence class (see also [3] for the specific case of the trefoil group). There are infinitely many such $T$-systems if $\vert m \vert,\vert n \vert \ge 2 $ and  $\vert m \vert + \vert n \vert > 4$.
But all these examples are again residually finite.

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[1] "Über die gruppen $A^{a}B^{b}=1$", O. Schreier, 1923.  
[2] "Über die Nielsensche Kürzungsmethode in freien Produkten mit Amalgam", H. Zieschang, 1970.  
[3] "Presentations of the trefoil group", M. J. Dunwwoody, A. Pietrowsky, 1973.    
[4] "Transitivity systems of some one-relator groups", A. M. Brunner, 1974.  
[5] "Generators of the free product with amalgamation of two infinite cyclic groups", H. Zieschang, 1977.  
[6] "Nielsen equivalence in closed surface groups", L. Louder, 2010, [arXiv:1009.0454][7].  
[7]: http://arxiv.org/abs/1604.08896