I recently completed Ultima VII: The Black Gate and The Forge of Virtue, so I was wondering if I should take a detour for Ultima Underworld, and play both of them and then play Serpent Isle. I'm asking this because even though Ultima Underworld I: The Stygian Abyss is a side story that takes place during the Age of Enlightment, I know that the second Underworld game tells what happens between The Black Gate and Serpent Isle. Does playing Ultima Underworld I and II before Serpent Isle make it more enjoyable? Serpent Isle sounds like an awesome game (I loved the Black Gate and Forge of Virtue) but I've also heard great things about the Underworld games.
I would like to hear people's opinions about these three games and the order in which they would play them.

If you do plan to play all of them eventually, it makes sense to play Underworld II between The Black Gate and Serpent Isle. Ultima Underworld is largely ignored by the other games.

Underworld I is a game I'm glad I played but it's long and while fun enough I probably won't play it again. I don't feel I'm missing anything from not playing it again. Like Savage Empire and Martian Dreams, it's a self-contained story that starts and ends within the game and no characters (from memory) appear in any other Ultima.

Underworld II was much more fun to me, probably in part due to it being part of the Ultima story. The levels were smaller and easier to navigate which made it more fun for me as well.

There aren't a lot of Underworld II references in Serpent Isle from memory so if you didn't feel like a dungeon crawl you could go straight to Serpent Isle and not feel like you're missing important plot points. I played Underworld II long after playing Serpent Isle and didn't feel disappointed that I missed out on playing them in order.

Underworld II doesn't continue the story of Underworld I at all so can be played without Underworld I if you prefer.

Ultima Underworld II does foreshadow a few things that will occur in The Serpent Isle, and plot-wise you receive a certain very important item that you start SI with as a result of UW2's plot. (This doesn't actually happen in-game in UW2 due to a glitch but SI refers to it happening; you can consider that the individuals gave it to you just after successfully completing UW2 if you prefer.)

The main reason to play it is that it more thoroughly explores the nature of the Guardian (at least as he was prior to the horrible U9 retcon) and his plans not just for Britannia, but for everywhere else he can get his red hands on. If you needed more reasons to loathe him after U7, you'll get those in UW2 before the plot events start to snowball in SI, which turns to his reaction to not getting what he wants (U7 and UW2) or even being handed setbacks (UW2).