Challenging to answer this generically for all biomolecules.

Biochemical systems are always in some kind of dynamic flux. Think about proteins on the surface of a cell. Those proteins are regularly turned over - old ones get internalized (then broken down into amino acids to be repurposed for new proteins), and newly synthesized proteins get trafficked to the cell surface. This turnover happens constantly and the time scale depends on the specific protein. However this generally happens on the order of hours to days.

The added complexity to this is that the conversion of one biomolecule to another often does not happen spontaneously, but is facilitated by enzymes. A protein suspended in some kind of tissue will likely degrade faster compared to it being suspended in pure water.

If we ignore enzymes, there are all sorts of factors that affect a compound's stability. Temperature, exposure to specific wavelengths of light, presence of water or dried as a powder, pH, salt concentration, etc. Note that many research chemicals (including proteins) are stored as powders because they are more stable in that form compared to being dissolved in solution. This all to say the lifetime of a given biomolecule depends on a variety of conditions.

To your main question - how long do biomolecules get passed around by nature? Potentially indefinitely and infinitely. In nature, it's unlikely that a given biomolecule will just be left alone to passively degrade into individual atoms. Another organism will likely do something with that biomass. That being said, we fight nature all the time to slow this down, ex: we keep meat on ice and covered in salt dry it out and slow down the decaying process. With enough effort and $$, you could certainly slow down the degradation of any compound to keep around longer.

There is a very cool experimental method where you make a version of a compound you care about but use radioactive isotopes for some of the atoms. If you treat cells in a dish with this compound, you can then measure where that radioactivity is and then infer what biochemical pathways it went through to get converted into something else. This is called a pulse-chase experiment (you pulse the system with the radio-labeled compound).

TL;DR - Biomolecules generally do not exist in a vacuum where they have a chance to passively decay. Barring specific laboratory settings, biomolecules are basically always part of some network of biochemical reactions and constantly being converted into other compounds.