Lore-wise, you won't miss anything because all important information will be explained to you (even if you never played the main base game story, when you come across Abnur Tharn you get plenty of lore explanation of who he is and why you should care about his character). The game also fills you in on events and other background info you might not be aware of (e.g. Euraxia taking over parts of Elsweyr through an Imperial power grab while the rest of Cyrodiil was busy with Varen's civil war), and characters that you come across tell you about themseves too even if you haven't met them.

However, personal story-wise it's more rewarding to play chronologically, because devs often write dialogue that acknowledges your previous deeds and acquaintances (e.g. Raz in Summerset will refer to your deeds in the base AD story if you did it, but Raz in the AD story won't refer to your Summerset deeds), so you lose out on a bit of flavour and custom dialogue if you play things out of order. It's also fun to see what your old pals have been up to since you last adventured together. And that prior experience can turn a forgettable side quest with a random NPC into a knowing reunion with a fond throwback character!

That said, play what you feel like playing in the order you want really, because there's no point in forcing yourself to play through a DLC you have no interest in just so a few characters throw you a line or two.