Yes, NPCs and creatures can drown.
Based from multiple UESP wiki articles and reports from various Oblivion forums, I found that NPCs and creatures that do not have the 'Water Breathing' ability can drown, and will go up for air once they're underwater (on creatures, this behavior will depend on their AI).
Among those that can't drown are aquatic creatures like slaughterfishes and its different varieties, Undead creatures and Argonian NPCs, which all have the 'Water Breathing' ability. Some Undead creatures have the 'Water Walking' ability, instead of 'Water Breathing'.
Some reports I found from UESP wiki articles (emphasis mine):
It is possible to go underwater with your horse, but be warned; horses can't breathe underwater and tend to end up in a watery grave. This can be avoided but it is tricky to get a horse back above the water.
From: UESP wiki, "Horses" (Oblivion) article
Undead cannot be drowned: they all have either Water Walking or Water Breathing as a permanent ability.
From: UESP wiki, "Undead" (Oblivion) article
It is also possible to drown guards, with Water Breathing and a relatively small bounty. To accomplish this, it is important to find a body of water with an obstruction directly above it, such as a dock in the Bravil Canals, or the Mage's Guild Well in Cheydinhal. Strike a guard, prompting them to follow you underwater. When the guard runs out of Breath, he will be unable to surface due to the obstruction directly above him, and will drown.
From: UESP wiki, "Guard Uniforms" (how to acquire City Guard uniforms without receiving a large bounty) article
Reports from various forum posts:
Yes they can [drown]. I was being chased by the guards in Bravil once, and I dove underwater and managed to trap one under a bridge. They do try to come up for water, but if something is blocking their way they will eventually run out of air and lose health. It's actually a fun way to kill people once you master it.
If the creature is not programmed to breathe underwater (like slaughterfish) then it will drown when forced underwater.
It is a lot more difficult though to drown creatures than NPCs.
From: this Reddit /r/Oblivion comment
NPCs can definitely drown. Many creatures can't actually swim, but the ones who can should also be able to drown. I don't think the AI allows for them to come up for air, unless it's coincidental. Usually the only reason an NPC or creature would be underwater is because their AI forces them to be.
From: this UESP 'Oblivion' forum post
...I'm sure I've drowned an [ogre] at least once by a combination of damage fatigue and paralyse spells.
It also might be worth mentioning that NPCs can drown. All land creatures except those with specific water breathing qualities and liches can be drowned.