Mountains and other geological features act just like obstacles do against waves: they make the waves bounce and reflect.
A nuclear blast has three primary effects that are detrimental to your health:
1a) The heat pulse. Immediately upon detonation, there is a very strong "light" coming from the bomb. It is like the Sun, only many times worse, depending on how close you are. This heat pulse can be enough to set flammable materials on fire, and to char other materials.
The front of the house, facing the intense light from the detonation, chars
But this heat pulse is stopped if you find yourself in the shade from it. This is why the now infamous civil defense movie "Duck & Cover" was made, because if you are on the fringes of the blast area you might save yourself from some serious damage if you get yourself into the shade quickly.
A mountain — obviously — would provide quite excellent shade.
Then again: a large mountain is may times larger than impact area of a nuclear bomb, even in the megaton range.
1b) The radiation pulse. At the detonation, there is also a pulse of radiation coming directly from the bomb. This is not quite as easy to protect yourself from because radiation like neutron and gamma are not stopped by being in the shade, but instead dampened.
However... if you have an entire mountain in the way, then that is enough to dampen the radiation into nothingness.
2) The blast wave. The blast wave is a sudden and very severe blast of wind. At close enough range, this blast pulverizes houses, sends vehicles flying, knocks over structures.
The blast wave has just hit the house
Again: a mountain would deflect this wave. If you are in the "shade" of a mountain, you'd be quite safe.
And yet again: the blast zone around a nuclear blast is smaller than a mountain like Mount Fuji.
Here it a video that shows the heat pulse and the blast wave very clearly. This is the Teapot Apple II shot from 1955. This is a "small" explosion at 29 kT.
3) Fallout. Fallout is the radioactive waste products of the blast. Fallout — unlike the direct pulse of radiation at the blast — do not have a very significant range. Think of it more like a big cloud of dust. If you see the cloud in the distance, then that is OK. What you do not want is for it to come down on top of you so you start breathing it.
Here things become a lot more complicated. A mountain will most certainly have an effect on fallout. Since mountains affect wind, rainfall and such it will have an effect on the fallout. A plume that comes towards the mountain and begins to climb over it may encounter rain and be washed out onto the mountain side instead of coming over to where you are.
Another aspect is important here for that which you said: your explosion is at ground level. This will make the fallout much more intense, because it rips up lots of dust that the radioactive fission products stick to. At 10 MT, getting caught in that fallout can be very dangerous. Compare the Japanese fishing boat Lucky Dragon 5 that got caught right in the plume after the 15 MT Castle Bravo shot. All crew members got acute radiation sickness, one died of secondary effects.
Finally there is the visual aspect that we all know when it comes to nuclear bombs: the mushroom cloud.
If we are talking about a 10 MT bomb, the mushroom cloud will rise to about 20-30 km in altitude so no mountain will be able to hide it. This picture...
...shows the size of mushroom clouds for different yields.
So to answer your question: anyone who is on the other side of the mountain from a 10 MT blast will notice what seems to be a sunrise on the ground on the other side of the mountain. They may notice a strange short-lived puff of wind and hear a distant rumbling after a while. And then they will see the extremely ominous mushroom cloud rising behind the mountain, high above it.
As for anyone on top of the mountain.... well it all depends. Do they have a direct line of sight to the blast? If so a 10 MT shot can cause anything from first to third degree burns. The blast wave can travel up the mountain side, so — again — if they have a clear line to the explosion epicenter, they might get hit pretty bad by the blast. But the outcome can be tailored however you want it by simply adding shade and cover.
Oh and for the record... all that talk in the comments of "Oh the mountain will be destroyed!"? That is complete hogwash. We are talking about a mountain versus heat, wind and radiation. Rock doesn't give a hoot and a holler about that. There will be a crater where the explosion happens but it will not be very big compared to the mountain. By comparison; the Castle Bravo shot at 15 MT left a crater that was 2000 meters wide, but only 76 meters deep. That is to say the depth was only about 1 in 30 compared to the diameter.
Since it was a ground burst, the fallout will be severe. If it affects the character in your story, all depends on where the wind is blowing. If it comes towards them you can place them in a lot of trouble. If the wind blows away, they will be safe unless they wander into the blast area.