LONDON— When Poland’s right-wing Prime Minister Beata Szydlo swept to an election victory late last year, one of the first things her government did was remove the blue European Union flag from the backdrop of her news conferences, and leave only the red and white Polish flag. It was a symbolic, patriotic gesture that signaled a more distant relationship with Brussels. At the same time in Austria, more than 260,000 citizens signed a petition that forced the Austrian parliament to open a discussion on a referendum to leave the European Union. Eurosceptics in both countries are now watching Britain’s march...