Density is 19.25 g/cm3, hardness ~9 (Tungsten carbide might apply), and melting point is at 3695 K.
It will never get past the mantle.
The tungsten (or tungsten carbide) will chemically react with the rocks surrounding it to dissolve and form other minerals (silicates), or even worse: get diluted to trace levels in mantle minerals. Compare this with dropping some grains of salt in the ocean and asking when they will arrive to the sea floor. Your block of tungsten will probably need to be on the scale of tens of metres or more (km?) to escape that.
Furthermore, the mantle is convecting so it will keep your thing from sinking directly down. Compare with leaves falling from a tree in a thunderstorm. This will encourage the chemical mixing I mentioned before.
Yes, tungsten carbide does have a hardness of 9, but neither it nor diamond, which has a hardness of 10, are immune to wear from friction. It's how rough diamonds get transformed into cut diamonds.
From personal experience of using tungsten carbide drill bits (chisel bits, cross bit & button bits) to drill holes in rock, the tungsten carbide wears out and needs continual sharpening. The harder the rock being drilled, the faster the rate of wear. Eventually each drill bit wears out & becomes so narrow that it can no longer be used and it must be replaced. Also, anyone who has had involvement with diamond drilling will confirm that diamond drill bits wear out when they drill rock.
In trying to reach the Earth's core, the hole will deviate and not be dead straight and the drill bits and drill steels will need to deal with high lateral earth pressures at depth and increasingly high rock temperatures. Additionally, tungsten carbide is had but brittle so it's uses are limited.
No-one yet has the technology to drill a hole into the Earth's mantle, let alone its core.
Edit: After the question was edited to clarify the tungsten spike would move by gravity alone.
Nothing much will happen. Such a spike, regardless of it mass, will sink into soil or sand, but once it contacts denser material, such as rock, and unavoidable lateral earth pressures also squeeze on the spike, movement of the spike will stop and it will still be near the surface of the Earth.
Gravity alone will not provide such a spike with the energy it needs to split rock.