Empathy is shaped by many factors: our genetics, our family history, our personalities, and our environment. It's also molded by the experiences we have with other people and whether or not we choose to put ourselves in their shoes and feel their feelings. Some people resist doing this because they've been so badly damaged themselves and don't want to take on more heartache. However, by doing this, they're shutting down their very humanity.
This was the case with my own mom who was rejected as a child by her alcoholic mother. When growing up, I was a sensitive kid who felt things intensely, but my mom never provided a safe spot for my emotions. In fact, she didn't know how to deal with them at all and typically reacted with anger and frustration.
Some people (perhaps, your mother and grandmother) make a conscious or unconscious decision to be unsympathetic. They even mistakenly see it as a sign of strength. After my son was diagnosed with autism, I decided to just focus on my immediate family and walled myself off from other people's problems. I thought if I took on another person's pain it would crush me. When I was ready to open myself up again, I had to re-train myself to be empathetic. There's no doubt it takes practice and patience. If we're too busy and overwhelmed with life, our empathy declines dramatically.
People in helping professions such as social workers, teachers, nurses, and therapists often become victims of “compassion fatigue.” To prevent this from happening, they must learn to recognize the pain of others but not let it become their own. They need to differentiate themselves from the person who's suffering. They must put themselves first and have personal lives in which they exercise, eat right, and have hobbies, friends, and fun times so they don't become burned out in their careers.
If you're experiencing empathy to the point of self-destruction, you need to take a step back and realize the negative impact it's having on your body and your brain. Don't lose yourself in other people's problems. Take time for yourself and surround yourself with positive folks.
You're an impressive person who's not allowed your mother's coldness to make you detached. You should be very proud of that but remember to put yourself first.