I have been told I have a grade 1 tear of the ligament of the right index finger, how long does it take to heal?

Varies. Typically a grade 1 means there is pain with minimal injury to the ligament. If there is an actual tear, the grade goes up to a grade 2 or 3. Healing varies per person. When there is full range of motion, no swelling, and equal strength compared to your opposite hand, then it is safe to return to sports.
Motion, compression. Grade 1 sprains cause no long term problems and generally heal in 4-6 weeks (or sooner). They heal best when the joint is kept moving. Treatment consists of motion, control of the swelling, and gentle compression around the joint.
"Heal" quickly. I use quotes because while the ligament "heals" quickly, the finger can remain swollen and stiff for a very long time. Sometimes it's just a matter of weeks; other times many months. Either way, the best treatment is to keep moving and stretching the finger joint.