Sounds like you’re doing really well, OP! I’ve been in your shoes, as have many others on Mumsnet, I’m sure, and know how tough it is.
If people ask if you’ve been kept awake, just smile and say something generic like ‘same old, same old!’ or ‘haha, I need my coffee today for sure!’ but say it cheerfully and then quickly move the conversation along. No need to lie or pretend you’re rested if you aren’t, but no need to discuss it either.
They will only get annoyed if you moan loads (although it doesn’t sound as though you’re a moaner!).
By the way, if you ever DO drop a ball, remember that you’re only human and mistakes happen far more than you realise - you just don’t hear about them as people don’t talk about it much. When my kids were young and I felt I was underperforming compared to my usual performance, I always used to remind myself that I was unlikely to be fired (it’s actually really hard to fire someone unless they are practically deliberate in their incompetence), so I just tried to relax a bit and learn to ride the wave of balancing parenthood with work!
Remember you’re not doing LESS than before, you’re doing an awful lot more - as you’ve working a dayshift at work and then a full night shift at home. I’d say that’s Superwoman skills! :-)
ps this definitely won’t last forever. You WILL sleep again