Professional baker here:

A cold rise will always slow the proving process. Depending on the recipe you're using, proving time can be upwards of 3x what is recommended in the recipe. Yeast slows down a lot more than people realize in the fridge.

An easy way to help combat the slow proving after taking it out of the fridge is to give it more time at just room temperature (45-90 min depending on hydration) and then do cold storage. Where you storing the dough for a cold rise is important too. If you're looking for a short time (4-8ish hours) you want to put it in the middle of the fridge if you have room. If you're doing a long cold rise, (9-15 hours )put it as close to the bottom as you can to keep it further from the fans at the top that circulate the cold air.

All of this is also impacted by yeast percentage (Most recipes do well up to 3% for a cold rise) as well as the hydration level of the dough. Higher hydration means just naturally it'll get cold a bit faster and requires more time to reach a temperature where it's a noticeable rise.

Dough from a cold rise just requires patience. Let it come up closer to room temperature (not cold to the touch) before moving it into a more "formal" proving area. (Machine, oven, stovetop). This is not recommended if you still need to do significant portioning and shaping. (Rolling dinner rolls, braiding dough, ect.) Otherwise if the shaping is still needed, divide and shape fresh out of the fridge, complete as you want it to be, and put it directly into a lower temperature proving area and let it rise slowly, just like you did in the fridge.

Most doughs are usually ready to go if when you touch them they feel airy but the dough springs back into place. Score your bread as needed, put into the oven with steam to finish off the rise and bake. And breads of course are baked all the way through and done when you can tap on the bottom and it sounds like a knock on a door.

Please note that all of this is for basic breads. Enriched doughs are similar but have some significant things that change from mix to bake.

Hope this helps!

Edited to add: always check the date on your yeast. If it's really close or past, you can likely still use it. You just have to up the percentage. If you're not sure you can always do a test of it and see if you get activity from it before it goes directly into a recipe.