Short answer: $t \lesssim 10^5\ \mathrm{years}$ (maybe)
An "overcontact binary" is just another way of saying "common envelope binary". The two phrases are exactly the same and it's frustrating that the authors on the VFTS 352 paper decided to create their own convention - as if astrophysical classifications weren't confusing enough!
A contact binary exists on timescales predominantly dependent on stellar evolution, so figuring out how long a contact binary will exist is heavily dependent on the mass, metallicity, and rotation of the primary star among other things.
Deriving the timescale:
Let's keep the scope to systems like VFTS 352, where the primary is massive and the binary has an orbital period less than 4 years (2.5 AU separation). In order to have a common envelope event, the stars must have overflown their Roche lobes. The radius for the Roche lobe of two point masses is
\begin{equation*}
r_L = \frac{0.49 q^{\frac{2}{3}}}{0.6q^{\frac{2}{3}}+\mathrm{ln}(1+q^{\frac{1}{3}})}a
\end{equation*}
where $a$ is the separation. For close binaries, the general observed trend is a high mass ratio $q=M_2/M_1$. So, if we assume $q=1$, then $r_L = 0.38a$. Hence, for a binary with $a<2.5$ AU,
\begin{align*}
r_L &\lesssim 1\ \mathrm{AU}\\
r_L &\lesssim 215\ R_{\odot}
\end{align*}
since $q=1$ is an upper bound on the Roche lobe radius. Now, performing some trivial rearrangement of the blackbody luminosity equation $L=4\pi\sigma_{SB}R^2T^4$, we find that
\begin{equation*}
R \approx 3.31\times10^{7} \bigg(\frac{L}{L_{\odot}}\bigg)^{\frac{1}{2}}\bigg(\frac{1\ \mathrm{K}}{T}\bigg)^2 \ R_{\odot}.
\end{equation*}
Massive stars typically have roughly constant luminosity, so we will choose $L\approx10^5\ L_{\odot}$. Hence,
\begin{equation*}
R\approx 1\times10^{10}\bigg(\frac{1\ \mathrm{K}}{T}\bigg)^2 \ R_{\odot}
\end{equation*}
The massive star needs to evolve until its radius is equal to that of the Roche lobe radius, so we find that the star reaches the common envelope phase for
\begin{equation*}
T \gtrsim 7000\ \mathrm{K}
\end{equation*}
Taking a peek at an HR diagram, this star varies from about $30000\ \mathrm{K}$ to $4000\ \mathrm{K}$ from ZAMS to end of main-sequence. Thus the primary spends roughly 3/4 of its time on the main-sequence not in the common envelope phase. Hence, this binary's common envelope phase lasts for, at most, 1/4 the primary's total lifetime, which is on the order of $10^6$ years. Thus, the upper bound for the timescale of a common envelope event with massive stars with negligible rotation is $\sim10^5$ years.
Please note that this derivation does not take into account the bulging effect that occurs as the separation decreases. This will certainly lower this upper bound, but by how much I'm not sure. It could lower it by 1 year, or $10^5\ \mathrm{years}$.
Lower bounds to this timescale are entirely ambiguous and not particularly helpful in any physical context. The stars could be spinning really fast, have high or low metallicity, the binary could have a different mass ratio, there could be another binary close by, and there may be magnetic interaction (?). The list goes on! I'm sure there's something I left out.
