In the book by P.G. de Gennes Superconductivity of Metals and Alloys, it's written
$N(0)V < 0.3$.
Also written :
Lead and mercury are two notable exceptions with low $\Theta_{D} \left( =\hslash \omega_{D}/k_{B} \right) $, giving, respectively, $N(0)V=0.39$ and $N(0)V=0.35$.
More details on p.112, P.G. de Gennes Superconductivity of Metals and Alloys, Westview (1999). The first edition dated back to 1966. To my knowledge there is no change between the editions.
See also the table 4-1 on p.125 of the same book for several specific values for the pure metals. This table is reproduced below for commodity.
$$\begin{array}{cccc}
\mbox{Metal} & \Theta_{D}\left(\mbox{K}\right) & T_{c}\left(\mbox{K}\right) & N\left(0\right)V\\
\mbox{Zn} & 235 & 0.9 & 0.18\\
\mbox{Cd} & 164 & 0.56 & 0.18\\
\mbox{Hg} & 70 & 4.16 & 0.35\\
\mbox{Al} & 365 & 1.2 & 0.18\\
\mbox{In} & 109 & 3.4 & 0.29\\
\mbox{Tl} & 100 & 2.4 & 0.27\\
\mbox{Sn} & 195 & 3.75 & 0.25\\
\mbox{Pb} & 96 & 7.22 & 0.39
\end{array}$$
Post-scriptum:
There are (surprisingly !) nothing about this question on the book by J.R. Schrieffer, Superconductivity, Benjamin (1964). There is not even a discussion on the gap equation as far as I can check... There is a repetition of the Gennes data on the book by A.I. Fetter and J.D. Walecka, Quantum theory of many-particle systems, Dover Publications (2003, first edition 1971), p.448. But this table is less complete.
There is also no discussion about the numerical value in the original paper by BCS [Bardeen, J., Cooper, L. N., & Schrieffer, J. R. ; Theory of Superconductivity. Physical Review, 108, 1175–1204 (1957). http://dx.doi.org/10.1103/PhysRev.108.1175 -> free to read on the APS website], but there is a possibly interesting expression (Eq.(2.40), written below in your notation / in the original BCS paper, the gap is written $\varepsilon_{0}$):
$$\Delta\left(T=0\right)=\frac{\hslash\omega_{D}}{\sinh \frac{1}{N(0)V}}$$
which might be of help for calculating the critical line $\Delta(T)$.
The expression you wrote is strange. It looks close to the current-phase relation for a ballistic and short junction:
$$j\left(\varphi\right)=2ev_{F}N_{0}\Delta\sin\dfrac{\varphi}{2}\tanh\left(\dfrac{\Delta}{2k_{B}T}\cos\dfrac{\varphi}{2}\right)$$
when $\varphi=\pi/2$, which is not the critical current at low temperatures, since $j_{c}=\max\left\{ j\left(\varphi\right)\right\} $. You can plot the above expression for different temperature (or large ratios $\Delta/2k_{B}T$ since it's simpler then) to realise that the current-phase relation becomes less and less sinusoidal at low temperatures.
Anyway, there is no reason why your junction should be ballistic, short, and the contacts with the superconductors should be perfect.
I didn't open it since a while, but the reference :
A. Golubov, M. Kupriyanov, and E. Il’ichev, The current-phase relation in Josephson junctions Rev. Mod. Phys. 76, 411 (2004)
might be of interest for you. It should be a discussion about the role of contacts, diffusive and ballistic limits, short vs. long junctions, etc...
Best Answer
This interpolation formula agrees with both the high- and zero-temperature limits for $\Delta(T)$: \begin{align} \tag{1}\label{eq1} 1-T/T_c \ll 1 &: \Delta(T)\approx 3.06\, k_B T_c\sqrt{1-T/T_c}\\ \tag{2}\label{eq2} T = 0 &: \Delta_0 = 1.764\, k_B T_c. \end{align}
When $T$ is near $T_c$ the argument of $\tanh$ in the interpolation formula is small so we can approximate $\tanh x \approx x$, giving $\Delta(T) \approx k \Delta_0 \sqrt{T_c/T-1}$. Then to recover \eqref{eq1}, we substitute $k = 1.74$, replace $\Delta_0$ using \eqref{eq2}, and use the fact that $\sqrt{T_c/T-1} = \sqrt{1-T/T_c}$ for $T$ near $T_c$.
At zero temperature, the argument of $\tanh$ in the interpolation function is large such that $\tanh x \approx 1$, giving $\Delta(T = 0) = \Delta_0$ which is just \eqref{eq2}.
As for the choice of $k$, for some strong-coupling superconductors, the prefactor in \eqref{eq2} is larger than 1.76.
Not only does the interpolation formula agree at $T=0$ and $T=T_c$, as stated in the comment above, but it also works near $T_c$.