There are 3 main methods of regression for panel data. Pooled OLS, Fixed and Random Effects.
Mainly to select Pooled OLS you'll need to test for individual effects in the error, this mean that Var(u) differs from 0 and E(u) is also different from 0. So there is presence of individual/ Heterogeneous effects in your OLS regression and your estimators will be biased. You can confirm this Breush-Pagan LM test. So you need a futher method of regression.
If such thing happen. then you'll need to select a fixed/random effect above your OLS estimation. and you do that with Hausmann test.
Under these conditions, Modelling data panel analysis has different points about the considerations of unitroots and cointegration. But I like to take what Park (2011) and similars say about the Modelling. That there is no test of unitroots, cointegration needed to model under the fixed/random effects.
However under different estimation methods, I believe it's better to test the unitroot/stationarity of the variables, morelikely if T tends to equal N. In this case spurios regression may outcome. A formal test of unitroot rejected means your data is stational, but if this hypthesis is not rejected, cointegration should be tested. And then regression methods would be a bit more accuracy.
The idea behind the testing of unitroots and cointegration derives from the assumptions of the regression model. Random effects for example according to Woolridge has 3 core assumptions, but noone of them are related to the time series unit root. except for the one where (u_i) shall not be correlate with (u_j) for any time period inclidung i=j, the cross sectional dependence may lead to biased estimators, so the common correlated effects method of regression would be better and there unitroot and cointegration should be tested.
As you see the Panel Data Analysys without including the time regression pattern of test is well detailed. So papers don't bother about including the stationarity or the unitroots presence over the random/fixed effects methods, not even Pooled OLS.
I'd like to stay with the basics from what Park says about modelling.
Park Myumng, Hun (2011) Practical Guides To Panel Data Modeling: A Step by Step. Analysis Using Stata. University of Japan.