WB, I had polyhydramnios with dd1. It was diagnosed at 38 weeks, via a scan - which was done because they thought I was having a huge baby (I wasn't, it was mostly fluid).
It might depend on the hospital, and any scans you've had, or initial APGAR tests etc but...dd wasn't whisked away at all. They did come to our room a few hours post birth and insert a nasogastric tube (I howled, she coughed once then seemed unbothered) which they left in for a little while, then removed. That was to check for GI blockages. Later she had a blood test, which came back fine except for slightly raised crp (inflammation marker) which could have meant something or (as it turned out) nothing. As a result, they wanted to keep an eye on her and give her antibiotics just in case - but she was fine.
So yes, I'd say it might delay you leaving the hospital by a day or two (if her initial tests aren't all clear, as dd's weren't) but it might not. Did they say they'd whisk her away? If so, why?
I know how I felt after being told about it, and I did'nt have the loss of a planned home/water birth nor the fear of PND....so can imagine this has come as a shock.
Seems like a few of us are having changes of plan at this stage...maybe shows how near the end bit we are?