Methods
This is a multivariate problem. In general, you will need arrays of $\sigma$, $\epsilon$, $\dot{\epsilon}$, and $t$ for the three values of $T$.
Method 1 - Fully Multivariate
The approach may depend on the code base, as the guidelines are likely different in each. You will likely need to create a function that accepts three arrays ($\epsilon$, $\dot{\epsilon}$, $t$) along with $T$ as inputs and returns the value of $\sigma$. You will need to run that function through the multivariate routine in the code base that you are using.
The advantage of this approach is that you fit all parameters globally. The disadvantage is only in your ability to construct and use the multivariate methods for the code base at hand.
Method 2 - Stepwise Regression
In this case, start by collapsing the temperature dependence.
$$\sigma= B(T)\cdot(\epsilon^{m_2})\cdot \exp\left(\frac{m_4}{\epsilon}\right)\cdot \dot{\epsilon}^{m_3}$$
Fit each of the three temperature curves separately to obtain the four fitting parameters $B(T), m_2 - m_4$. Then, fit the three values of $B(T)$ versus $T$ to obtain $m_1$.
The advantage of this approach is that it lends itself to an easier coding algorithm than Method 1. The disadvantage is that you treat the $m_2 - m_4$ parameters as though they are also temperature dependent when they may not be.
Method 3 - Attempt at First Guess
Take three points of $\sigma$ versus $\dot{\epsilon}$ (one pair each from each curves for $T$) where $\epsilon = 1$ to write
$$\sigma= A \cdot e^{m_1\cdot T}\cdot \exp\left(m_4\right)\cdot \dot{\epsilon}^{m_3}$$
With only three data points and four unknowns, this cannot be solved.
Take three points of $\sigma$ versus $\epsilon$ where $\dot{\epsilon} = 1$ to write
$$\sigma= A \cdot e^{m_1\cdot T}\cdot(\epsilon^{m_2})\cdot \exp\left(\frac{m_4}{\epsilon}\right)$$
Again, with only three data points and four unknowns, this cannot be solved.
The combination of the two equations with six data points has five unknowns. The set can be solved. The approach goes back to Method 1. So, just do Method 1 on the entire data.
Caveats
As with any regression fitting of a theoretical expression to discreet data, the consistency (precision/repeatability) and accuracy will depend on having two things: an appropriate span of data points and a robust regression algorithm.
For this case, the theoretical curve is non-linear in the values. Given the exponential nature of all fitting parameters (except $A$), data should likely be measured over at least two if not three orders of magnitude of the parameters ($\sigma$, $\epsilon$) and over at least four values of $T$ across a range where the exponential dependence is anticipated to be well-resolved over just a mildly linear variation. An additional complexity in this situation is that $\dot{\epsilon}$ is not measured directly, it is obtained by derivation (differentiation) from $\epsilon$ versus $t$. Some benefit may be obtained by generating an analytical expression for $\dot{\epsilon}$ versus $\epsilon$ rather than a data array that is obtained from numerical differentiation.
The robustness of the algorithm may be limited by the application used. The common chi-squared minimization used in a Levenberg-Marquardt method may not be the fastest or most reliable compared to an entropy minimization method for example. In any case, until the span of data is appropriate, even the best algorithm is bound to fail more often than it succeeds.