Is the road from Delhi to Nepal Safe?

I imagine the new bus service would follow the Narayanghat Mugling Highway to get into the higher valleys and onto Kathmandu or Pokhara.

Without knowing where you went within Nepal 20 years back, I would hazard a guess that maybe you went up either the Siddharta Highway (to Pokhara) or the Tribhuvan Highway (to Kathmandu). Both of these roads are still subject to frequent closures due to washouts from monsoon rains and rock slides.

Is it any safer/better today? Rock slides and wash outs can occur even today, as mother nature is stronger than most everything man can build. If you are not traveling during the rainy season, then likely the ride will go through more or less on schedule. If you need to go during the rainy season be prepared.