I know once breast milk is warmed up you need to use it within 2 hours but also know that is with you actually warming it up in hot water. So far I can't find anything about letting it gradually warm up to room temperature without using hot water.
The CDC has some useful guidelines on this. If you thaw breast milk in the fridge, you can use it within the next 24 hours. Once it is warmed or brought up to room temperature (no matter the method), you should use it within 2 hours. I hope that helps!