The techniques are similar to the answer here Small noise limits with irregular drift.

In the article "On stochastic differential equations with locally unbounded drift" they obtain a beautiful pointwise almost everywhere result for the coefficients that then gives uniform. So on that random set you have, we only need that eventually it converges to $\sigma$ almost everywhere.

The only caveat is that you need assume uniform ellipticity for the volatility coefficient.

Then they get nice uniform convergence results

The techniques are similar to the answer here Small noise limits with irregular drift.

In the article "On stochastic differential equations with locally unbounded drift" they obtain a beautiful pointwise almost everywhere result for the coefficients that then gives uniform. So on that random set you have, we only need that eventually it converges to $\sigma$ almost everywhere.

The only caveat is that you need to assume uniform ellipticity for the volatility coefficient.

Then they get nice uniform convergence results.