This would rule out the many mosques that are in currently majority Christian countries in the Balkans that were built under the Ottoman Empire, as well as the mosques built in Iberia prior to the reconquista.

Malta had a majority muslim population under Christian rule after the Norman conquest of the island in the late 11th century until 1249 when all free muslims were expelled. However, I cannot find a record of a new mosque being built in that time.

Perhaps a better bet would be a mosque built by Lipka Tatars in the Grand Duchy of Lithuania. The Lipka Tatars settled there in 1397 after being given refuge by Vytautas the Great, though settlements of captured Lipka Tatars had been in the country since the mid-14th century. I can find a record of mosques being built in the Grand Duchy of Lithuania (modern-day Lithuania/Belarus/Ukraine) by the 16th century--a document from 1558 mentions a mosque--though I cannot confirm exact date of building or locations. I find it hard to believe that the Lipka Tatars, who were given the privelige of religious freedom in exchange for their military service, didn't have places of worship as early as the 15th century in the Grand Duchy of Lithuania, though of course this may not have included any purpose-built mosques.

In a similar timeframe, the Qasim Khanate, vassals to Christian Russia, built a mosque in Kasimov in European Russia in the mid-16th century, and the minaret still stands today.

The Shah Jehan Mosque, built in 1889 in Woking near London, claims to be the oldest purpose-built Mosque in the UK and "probably in Western Europe".

Sources:

Malta, Mediterranean Bridge. Stefan Goodwin (2002)

Muslims in Poland and Eastern Europe: Widening the European Discourse on Islam. ed. Katarzyna Górak-Sosnowska (2011)

http://www.wokingmuslim.org/pers/dr_leitner.htm

In the 1220s, Frederick II embarked on a program of forcibly transporting the Muslim population of Sicily to Lucera on the Italian mainland. It was part of an ongoing effort to preserve a Muslim population as Latin subordinates, not just to suppress rebellions. In Lucera, Muslims farmed very fertile fields (after all, Frederick wanted the benefits of taxation!) and were even allowed/encouraged to serve in Frederick's army. Julie Taylor even found a couple cases of Muslim soldiers who were knighted, owned land, and were exempt from paying the extra tax levied on non-Christians.

The Muslims of Lucera were allowed religious freedom and autonomy. Because they built their community from literally the ground up, we can assume that at least one mosque was built. Although there are few sources about religious life within the community, scattered references to gymnasia even suggest the presence of a formal madrasa or even a Quranic school that would have been attached to the mosque.

In 1274, the Muslim community of Huesca (Spain) was granted permission to quarry stones from old cemeteries for mosque construction, at the same time the local Dominicans received permission to quarry for a church. The source is not clear on whether "construction" means a new mosque, an expansion of an existing one, or simple repairs. We can't exclude construction of a new mosque: although the Church had banned building new synagogues (and by implication, mosques) earlier in the 13th century, the Spanish kingdoms continued to allow Jewish communities to build new synagogues.

Sources: Julie Taylor, Muslims in Medieval Italy: The Colony at Lucera; Alex Metcalfe, The Muslims of Medieval Italy; Brian Catlos, The Victors and the Vanquished: Christians and Muslims of Catalonia and Aragon, 1050-1300