The Paladin's Lay on Hands feature says:

As an action, you can touch a creature and [...] restore a number of hit points to that creature.

It also goes on to say:

Alternatively, you can expend 5 hit points from your pool of healing to cure the target of one disease or neutralize one poison affecting it. You can cure multiple diseases and neutralize multiple poisons with a single use of Lay on Hands [...]

Could I opt to do both at the same time? Perhaps an ally was struck by a crossbow bolt with drow poison, and I needed to both heal them as well as remove the poison in one go; is that permissible?

As written, it seems vague. The first paragraph specifies the use of an action to heal hit points, but the second simply indicates an alternative way for the hit point pool to be expended, but doesn't seem to indicate that the use is exclusive from the first.

Unfortunately not.

RAW, the use of Lay on Hands is heal HP, or cure disease. The keyword in the phrase is "alternatively" (emphasis mine).

Alternatively, you can expend 5 hit points from your pool of healing to cure the target of one disease or neutralize one poison affecting it.

The idea is that in combat, the process of using Lay on Hands only allows for you to either "heal" (HP), or focus more specifically on "curing" (disease), and this process is a "6 second" process.

Outside of combat however, this is less of an issue, so long as you manage your Healing pool points correctly.

However, Jeremy Crawford has clarified that it is intended to heal, and cure, at the same time.

No, not in the ordinary sense of "alternatively"

The semantics of alternatively, from Princeton Univerity's WordNet (which lists synonym sets together) are:

alternatively: as an alternative, instead, or else (in place of, or as an alternative to; "Felix became a herpetologist instead"; "alternatively we could buy a used car")

So notice that "alternatively" is listed as synonymous with "instead" and "or else".

Thus I do not believe it really is vague. If you neutralize a poison or cure a disease, you do so instead of restoring hit points.

According to Jeremy Crawford, yes, it can.

Official 5e rules designer Jeremy Crawford addresses this question quite simply here:

So yeah. Lay on Hands. Can you both heal HP *and* cure a disease in one action? I think yes.

Yes.

I think you can literally interpret the use of "alternatively" here in two ways, but in context only one of them makes any sense. You have, essentially, "As an action do a or do b". So it either is "(As an action do a) or (do b)", or "As an action (do a or do b)". If it's the first one, then that would suggest you could do b spontaneously without using an action, which I think in the context of the game and this ability doesn't make sense. So it would have to be the second one, which is as one action do one of two things.

Update:
Check out this answer: https://rpg.stackexchange.com/a/98393/50316. Notice where he quotes "you want to use a skill above-and-beyond passive use? You've got to devote some time and attention to it, as represented by your action." I'd say since Lay On Hands is clearly an active effort, not a passive effect, and would certainly take time and attention, it would definitely be an action.