Europe’s medicines watchdog is bracing to make do without one of its most trusted partners.

The European Medicines Agency stopped assigning work to the U.K.’s pharmaceutical regulator earlier this year, unsure whether the two organizations will need to stop collaborating after Britain leaves the European Union in March. Preparing for a potential split, the EU body has also handed over responsibility for about 370 drugs that were overseen by the British authority to other countries.

The EMA already stands to...