Agreed with lilacsigil that the cotton would be unrealistic, and adding on top of that statement that cotton is a very poor insulator compared to wool, as when it gets wet it cools the body off. Linen is a more popular liner for this reason, and an enduring one as a base layer rather than something specifically stitched into an outerwear garment like a coat - lots and lots of layers were preferred, because of ease of repairing garments, layering according to season and weather, and availability of various materials.

Additionally, coats by themselves are a bit wiggly of a preferred garment around the 1300s, particularly because they often first debuted as part of a military uniform and militaries themselves fluctuated based on need. Would your setting have a draft, or would it be primarily nobility with some enlisted footsoldiers, or some sort of mix of the two? This would inform who has access to a garment like that.

Fur would be a fairly traditional - and practical - liner, not least because of its warmth but also for the ease of hunting smaller game and stitching pelts together for a whole lining. Cloaks would also be hugely popular around the 1300s (and before, and also in some iteration afterward, now somewhat coming back into fashion in the modern day), because they're easy to make, easy to size (add belt!), and create a very warm bubble of air around you that coats with their sleeves are sometimes unable to mimic.