BRUSSELS — The European Union exported 25 million doses of vaccines produced in its territory last month to 31 countries around the world, with Britain and Canada the top destinations, just as the bloc saw its own supply cut drastically by pharmaceutical companies, slowing down vaccination efforts and stoking a political crisis at home.

As states have begun vaccinating Americans with medical conditions that may raise their risk for a severe case of Covid-19, they are setting widely varying rules about which conditions to prioritize.

The morass of guidelines has set off a free-for-all among people with underlying health problems like cancer or Type 2 diabetes to persuade state health and political officials to add particular conditions to an evolving vaccine priority list.