Yes, adopting the Euro in Britain would increase the inflation rate in the United Kingdom, because the pound is not as volatile as the Euro. The Euro is subject to the swings of a few volatile member states. While the Euro has a few strong countries, like Germany, to keep it going, the United Kingdom would be brought down by economic instabilities in other countries, which would result in inflation.
As with any shock to the system that is artificial and no based on reality, a temporary run would occur due to adjustable and partial irrationality. On top of this, the people at the time would try to hoarde as much protection, regulation, and money as they could while the panic when on, and then it would subside.