Elaborating on Deane Yang's answer and Willie Wong's comment: Since
$M^{2}\subset\mathbb{R}^{2}$ is a $C^{\infty}$ submanifold with boundary, the
Euclidean coordinates are global. Generally, if $M^{n}$ is a compact manifold
with boundary, we can cover it by a finite number of charts $\{x^{i}\}$, where
for any $C^{\infty}$ metric $g$ the functions $g^{ij}$ and $\partial^{\alpha
}g_{ij}$ are bounded (depending on $g$ and $|\alpha|$) and where $\alpha$ is a
multi-index with $|\alpha|\geq0$.

The scalar curvature $R_{g}$ (twice the Gauss curvature $K$ if $n=2$) is
$$
R_{g}=g^{jk}(\partial_{\ell}\Gamma_{jk}^{\ell}-\partial_{j}\Gamma_{\ell
k}^{\ell}+\Gamma_{jk}^{p}\Gamma_{\ell p}^{\ell}-\Gamma_{\ell k}^{p}\Gamma
_{jp}^{\ell})=(g^{-1})^{2}\ast\partial^{2}g+(g^{-1})^{3}\ast(\partial g)^{2}
$$
since the Christoffel symbols have the form $\Gamma=g^{-1}\ast\partial g$,
where $\partial^{k}g$ denotes some $k$-th partial derivative of $g_{ij}$ and
where $\ast$ denotes a linear combination of products while summing over
repeated indices. From the formula for $R$ we have for metrics $g,g^{\prime}$,
\begin{align*}
& |R_{g}(x)-R_{g^{\prime}}(x)|\\
& \leq C(|g^{-1}|^{2}+|g^{\prime-1}|^{2})|\partial^{2}g-\partial^{2}g^{\prime
}|+C(|g^{-1}|^{4}+|g^{\prime-1}|^{4})(|\partial g|^{2}+|\partial g^{\prime
}|^{2})|g-g^{\prime}|\\
& +C(|g^{-1}|^{3}+|g^{\prime-1}|^{3})\{(|\partial^{2}g|+|\partial^{2}
g^{\prime}|)|g-g^{\prime}|+(|\partial g|+|\partial g^{\prime}|)\left\vert
\partial g-\partial g^{\prime}\right\vert \}
\end{align*}
since $|g^{-1}-g^{\prime-1}|\leq C(|g^{-1}|^{2}+|g^{\prime-1}|^{2}
)|g-g^{\prime}|$.

Let $\hat{\Omega}=C^{2}(M,\operatorname{Sym})$. Given $h\in\hat{\Omega}$,
define $||h||=\sup_{x\in M}\max_{i,j,k,\ell}\{|h_{ij}(x)|,|h_{ij,k}
(x)|,|h_{ij,k\ell}(x)|\}$. Then $|R_{g}(x)-R_{g^{\prime}}(x)|\leq
C||g-g^{\prime}||$, where $C$ depends on bounds on the inverses and the first
and second derivatives of $g$ and $g^{\prime}$.

Elaborating on Terence Tao's answer and Deane Yang's comment: One reason it is
convenient to compute in local coordinates $\{x^{i}\}$ is that $[\partial
_{i},\partial_{j}]=0$. So the expression for the Christoffel symbols has only
$3$ terms instead of the $6$ terms comprising the formula for $\nabla$:
$\Gamma_{ij}^{k}=\frac{1}{2}g^{k\ell}(\partial_{i}g_{j\ell}+\partial
_{j}g_{i\ell}-\partial_{\ell}g_{ij})$, which is symmetric in $i$ and $j$. With
$\frac{\partial}{\partial s}g_{ij}=v_{ij}$, the variation formula is easy to
compute: $\frac{\partial}{\partial s}\Gamma_{ij}^{k}=\frac{1}{2}g^{k\ell
}(\nabla_{i}v_{j\ell}+\nabla_{j}v_{i\ell}-\nabla_{\ell}v_{ij})$, since the
computation of this tensor formula at any point $p$ may be done in coordinates
where $\partial_{i}g_{jk}(p)=0$ (such as normal coordinates centered at $p$);
this enables us to convert $\partial_{i}$ to $\nabla_{i}$ and to ignore the
$\frac{\partial}{\partial s}g^{k\ell}$ term since it is multiplied by terms of
the form $\partial g$. Now the variation of the Riemann curvature tensor is
$\frac{\partial}{\partial s}R_{ijk}^{\ell}=\nabla_{i}(\dfrac{\partial
}{\partial s}\Gamma_{jk}^{\ell})-\nabla_{j}(\dfrac{\partial}{\partial s}
\Gamma_{ik}^{\ell})$ using the same trick of computing at the center $p$ of
normal coordinates and replacing $\partial$ by $\nabla$ (note that
$\frac{\partial}{\partial s}(\Gamma\ast\Gamma)=0$ at $p$ by the product rule);
the resulting formula is true in any coordinates since it is tensorial.

Generally, it is convenient to compute in local coordinates because it can be
done more or less mechanically. For example, if $\alpha$ is a $1$-form, then
$\nabla_{i}\nabla_{j}\alpha_{k}-\nabla_{j}\nabla_{i}\alpha_{k}=-R_{ijk}^{\ell
}\alpha_{\ell}$. One can remember this as the contraction of
$-\operatorname{Rm}$ and $\alpha$, where the lower indices $i,j,k$ on
$\operatorname{Rm}$ appear in the same order as the first term on the left.
Similarly, if $\beta$ is a $2$-tensor, then $\nabla_{i}\nabla_{j}\beta_{k\ell
}-\nabla_{j}\nabla_{i}\beta_{k\ell}=-R_{ijk}^{m}\beta_{m\ell}-R_{ij\ell}%
^{m}\beta_{km}$, where the the lower indices of $\operatorname{Rm}$ are $i,j$
and then either $k$ or $\ell$, with upper dummy index $m$ on
$\operatorname{Rm}$ also replacing either $k$ or $\ell$ on $\beta$.