Then don't have a 'party.'
Their health and upcoming surgery...is more important.
And this is Flu season.
And cold season.
And if lots of people are around them, they may get sick or not. You don't know. People can also just carry a virus germ on them. And they get sick.
At 1 years old, they will not 'know' if they had a party or not.
You can just, you and Hubby, get them a cake and a candle and balloons for the house, and just as a nuclear family, 'celebrate' and take photos.
Then later when they are all recovered from surgery and are in the clear... you can have a belated party, if you wish. With everyone.
It is understandable.
And, I would not take them to Thanksgiving gatherings... either. Not with their upcoming surgery looming. The Docs told you... they must not get sick, if you can avoid it.
That is what I would, do.
If that were my kids. And they had a surgery coming up.
The priority here is their surgery. Right now. Leading up to it and after per recovery.
It is not easy, to reschedule a surgery. Docs go on vacation, too etc.
As you said, you cannot let them, get sick.
As for the cake, just get a cute little round cake with their names on it etc.
Put a candle on it for each of them.
My kids at that age, did not care about a cake, nor did they even care about eating it.
It is more for the adults and taking photos of it, with the child.
A 'sheet' cake, is too big. Especially since it sounds like, you are not having a birthday 'party' at this point.
But yet, are you doing a Thanksgiving party?? Since you mentioned that "there will be all sorts of Thanksgiving desserts and such...."
Sorry, it was confusing.
Friends of mine who have multiples, get 1 cake. With their child's names on it. And writing on it that states the birthday age. ie: "Happy 1st Birthday Sally and Johnny."