Yes, the lakes sit a short distance beyond the shrine and pair naturally with a Covadonga visit. Access rules on the mountain lake road can change seasonally, and at busier times a shuttle may be required, so it's worth checking current restrictions before you go. A private driver can plan around whatever rules are in place and get you to Covadonga and the lakes without you having to manage the logistics yourself.
Covadonga is about 83 km (52 miles) from Oviedo and roughly 82 km (51 miles) from Gijon, generally around one to one and a half hours by car depending on traffic and the mountain route. There's no direct train service, so a private transfer is the most comfortable way to reach the shrine, letting you enjoy the winding Picos de Europa scenery without watching a bus timetable.
Most travelers spend one to two hours at the shrine itself, enough time to see the basilica, walk down to the cave, and visit the fountain of the seven spouts. If you want to continue on to the Covadonga Lakes afterward, plan for a half or full day so you're not rushed on the mountain road.
Covadonga remains a working sanctuary and pilgrimage site, so modest dress and a quiet, respectful manner are appreciated, especially inside the basilica and cave. The path down to the cave involves a staircase, and the site can get busy with pilgrims and visitors alike, so a little patience goes a long way.
The 18th century neo-Romanesque basilica houses notable artwork and overlooks the valley. A staircase leads down to the cave, an almost mythical space holding the statue of La Santina, the tombs of Pelayo and Alfonso I, and a spring that feeds a fountain with seven spouts. Set against the Picos de Europa, the whole site has a striking natural backdrop.
In 722 AD, King Pelayo's victory here is traditionally marked as the start of the 700-year Reconquest of the Iberian Peninsula. The site has been a place of pilgrimage for the people of Asturias ever since, and Pelayo's tomb, along with that of King Alfonso I, still rests inside the sacred cave.