This might be too elementary for this site, but I asked first on math.stackexchange and didn't get an answer even after offering 250 bounty points.

Let $M$ be a smooth $m$-manifold and $N_1$, $N_2$ smooth embedded $k$-submanifolds such that $N_1\cap N_2=\partial N_1=\partial N_2$, $TN_1|_N=T N_2|_N$ and for $x\in N_1\cap N_2$, each vector in $T_x N_1$ is inwards-pointing wrt. $N_1$ iff it is outwards pointing wrt. $N_2$ and vice versa.

I would like to glue $N_1, N_2$ to one manifold such as $N_1\cup N_2$; unfortunately, this doesn't need to be a smooth submanifold of $M$ (think of the graph of $x^2\,\mathrm{sgn}(x)$ in $\mathbb{R}^2$).

Can I "change" $N_1\cup N_2$ in some arbitrary small neighborhood of $N_1\cap N_2$ to make the resulting manifold smooth? (Moreover, if I have a framing of the normal bundles on $N_1, N_2$ that coincides on $N_1\cap N_2$, can I have a smooth framing on the resulting glued submanifold?)

I tried to extend a nowhere zero vector field on $N:=N_1\cap N_2$ tangent to $N_{12}$ that points inwards wrt. $N_1$, extend it smoothly to vector fields on $N_1$ and $N_2$ and use the flow to construct collar neighborhoods $C_1$, $C_2$ of $N$ in $N_1, N_2$ diffeomorphic to $N\times [0,1)$ resp. $N\times (-1,0]$. Then I believe that the embedding $N\times (-1,1)\to M$ is a $C^1$ embedding. Is there a theorem that such $C^1$ embedding can be approximated arbitrary precisely (in Whitney topology) by a $C^\infty$ embedding that coincides with the prescribed one on $N\times ((-1,-1+\epsilon)\cup (1-\epsilon, 1))$ or something like that?