A short answer is "not significantly", assuming you just store your drink for the day, and there is no sharp changes in storage conditions (strong heat, freeze, etc.)---except warming up the amount you want to drink.
Coffee drinks are unstable, though, and a simple, freshly passed drip coffee tastes different (quality changes) with time. So quality will actually change over a day, although the difference is hard to notice (except people with very acute taste sense).
You are also adding milk, which softens the overall taste, so I think noticing changes in the coffee flavour is even harder.
Every vending machine around me (every 100m or so) sells coffee drinks in PET bottles. These bottles are stored for several weeks, and they sell pretty well. They taste fine for that price point and quality, and they are very stable. Of course, they do not include only coffee. They include milk, sugar, and hard-to-spell things. The last category does contain taste stabilizer, etc., so that is unfair comparison. I am just thinking it is interesting to see that coffee in plastic bottles in actually pretty common, especially for relatively short times.