Okay so - from what I can tell there is a global cooldown cycle split into two parts of a wave in ESO's combat. You can use a basic attack instantly after a skill, and a skill instantly after a basic attack clearing room for your basic attack cooldown and your global skill cooldown.
Weapon swapping counts as either or - so in a cycle of skill-light attack - skill - light attack ad infinitum one can swap out either a skill or a light attack for a weapon swap. If you try and weapon swap DURING a skill, light attack, while blocking or roll dodging the game simply won't allow it.
On another note - if you weapon swapped and you know you weren't doing any of the listed cycle actions but it lags - don't hit the button again, just use the hotkeys correlating to the abilities you expected to have after the swap - HEY PRESTO THEY WORK. It's just a small animation bug - nothing to worry about - the game will catch up soon enough.