I don't know if I had proper polyhydramniosis, but they did test me for gestational diabetese in 1st pregnancy as I was so huge, turned out to have been a LOT of fluid (showed on 35 week scan and flooded the place when they broke my waters at induction).
Baby was 9lb 6oz, fairly big though not huge. They never seemed worried about cord prolapse for me to be fair so I'm guessing I wasnt in the serious range water-wise!
However if it helps, they did let me go 10 days post due date, and I was induced. I would try to relax about it. For now all it probably means is you feel extra big and awkward (with all that goes with that, aches and lack of sleep etc). It may be comforting to know that the medical staff have it spotted and are being cautious, but you are likely to have a normal labour.
They may bring you in if they feel another 10 days of baby gaining weight isnt good. But induction shouldnt mean everything is out of your hands. Ask them for time in between pessary and waters being broken for example if you think your labour might start itself. Or if they break your waters and contractions startm, ask them to hold off on the syntocin (which I fould made contractions far more painful). You may get induced, but its still your labour. Let them know you want the bext chance to continue naturally even if they initiate things.
Whenever your water breaks, your midwife will know you have a lot of fluid, (you can let each midwife shift know if it makes you feel better) and they will be on the lookout for the cord. I cant see excess fluid as having any other effect on your birth.
As for pushing out a big baby, its all about the head size, extra chubbiness isnt really the thing. You may have a 10lb baby with a normal head circumference, or an 8lb baby with a large crown - I'd rather birth the 1st! Push like you are passing the biggest poo of your life, dont clench down there, focus your efforts and your baby has every chance of coming out the same as any other.
If for some reason you need vacuum, forceps or caesarean, its probably no link to the polyhydramnios, and at the end of the day the important thing is to have baby out safe. Induction at 40 weeks may actually give you a better chance of vaginal birth than letting you go 40+10 to a bigger baby and ending up needing assisted delivery or C section, so there may be a positive note if they bring you in early...?
Sorry for rambling, just rang a bell with me. Your worrying wont effect the baby, but it will effect you for these last weeks of your pregnancy. Try to relax, the high fluid has been spotted so will be watched, and when baby is out labour (good bad or indifferent) is behind you. Hope all goes well.