Hey OP, I work in obstetric theatres scrub side, and we really have seen it all. But we always try to preserve privacy and decency where possible, and any time you are exposed will be for a good reason, and we try to mitigate it as best we can.
So since you mention your lady parts, I'll list the times they will be exposed.
When you enter theatre, and before the spinal is inserted we'd ask you to remove your pants if you're wearing your own. If you're wearing net knickers or disposable pants, then we can remove them (i.e. cut them off) just before the catheter is inserted.
The first time will be when the catheter is inserted. We do this to every patient having a C-section. We can hold your gown up so you and your husband can't see what's going on, but obviously the nurse/ODP/midwife inserting the catheter has to be able to see what they are doing. Once it's inserted you'll be covered back up asap.
The catheter is necessary as it keeps your bladder drained, and thus as small as can be so is safely out of the way of the uterus reducing the chance of accidental bladder damage. It also gives us a way to see if there is any blood in your urine, which is an indication of damage. It's also the way we can test with dye to be sure there's no damage done, so it is vital.
The next time is cleaning. Unfortunately there's not much we can do here to mitigate this, as we need to area to be sterile, which means we cannot touch or cover it. Best we can do is to lift your gown again, so that you can't see, but at this point the entire team will be in theatre getting ready. It takes a minute or two total to apply the solution ensuring it covers everywhere (bump, lady parts, top part of your legs), and then for it to dry. Once it's dried, the drape goes on , and you're completely covered then.
Final time is post procedure. The lower portion of the drape is removed, with the upper portion left in situ so you can't see. You're frog legged (as when the catheter is inserted) and the surgeon removes any clots in your vagina, and checks there's no ongoing bleeding. Once that's done, a pad is put in place, your legs straightened back out, gown pulled down and/or you're covered with a sheet.
That it for theatre. In recovery, the nurse would like to check your pad every so often for bleeding. But this takes a couple of seconds, you'll still be covered by blankets, and the privacy curtains drawn.
Hope it helps, but for now, talk with your midwife about what will happen as well.