If you do something religiously, you can be relied on to do it on a regular basis. If you never forget to floss your teeth at night, no matter how tired you are, you can say you floss religiously.
You might go to church religiously, every single Sunday, but an activity doesn't need to be religious for you to do it religiously. Religion is the key to this adverb's origin — it originally meant just "in a religious manner." But these days you can use it to describe any activity you're devoted to, whether it's religiously watching your favorite TV show or drinking a cup of tea religiously every afternoon.