It won't take months. US will not follow Sweden. The shares on both exchanges will trade within pennies of each other immediately and always.
Interactive Brokers trades foreign exchanges, and many other brokerages I'm sure. I would not be the only one to short Sweden, buy US from day one, if Sweden was a p/e of 4 and US a p/e of 3.
That will never happen. I'd rather that it would, as it is free money.