Performance implications of larger UI textures?

When creating textures that I create sprites from for UI development, I typically use multiple 512x512 textures. A new UI design requires components that are larger than 512 (and one is larger than 1024).

What are the performance implications of using sprites from 1k x 2k (or 2k x 2k) textures for a UI. Not the HUD which will be displayed always over the game world, but a full screen UI.

My knee jerk reaction is to break everything up to fit on 512x512, even if it means creating larger images by compositing a few smaller; but then I haven’t really had to consider texture sizes in many years.