= Solution
The base <Riemannian metrics> satisfy $h_k\to h_0$ in the smooth topology, and the fixed smooth <covering maps> pull this convergence back to both $N_i$. Although the base has cone points, each lifted chart is an ordinary smooth chart, so convergence holds there as well. The fixed <isometry> $F_0$ used to mark the second limit gives
$$
\boxed{g_{1k}\longrightarrow g,\qquad g_{2k}\longrightarrow g\quad\hbox{in }C^\infty(N).}
$$
The area normalizations multiply by constants tending to one and therefore do not change these limits. There are precisely two sequences, indexed by $i=1,2$; the appearance of $i=1,2,3,\ldots$ in the printed last condition is a redundant indexing error, not a requirement for additional families. The common limit is isometric even though every finite pair is nonisometric, because the hyperbolic limit recovers the reflection symmetry broken by the locally rigid perturbations.
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