Time lapse between fresh blood's exposure to air and that same blood turning darker red-brown
In comments it been discussed that clotting is partial answer to OP. I would say, that is is somewhat tangential to the original question.

If I may re-phrase, OP asks: how one can estimate age of blood that has been exposed to air? Problem is that you will have to have a starting point to solve that backwards, like having an initial condition for solving differential equation.

One example that can be used is measurement of $pCO_2$ content in blood serum. If it is available, someone already collected this data. In 2 hours $pCO_2$drops 30% and total $CO_2$ drops about 6%.

Moreover, it seems that during hemorrhagic shock RBC shape changes, which means that sampling patient you will be able to approximate time since bleeding started. However, you will need an electron microscope at your disposal...

Finally, and I'll leave it like that, cellular death is a powerful process that leaves a lot of traces. Apoptotic cells have mitochondria bursting (except in RBC there are no mitochondria) and releasing, for example, cytochrome C. In case of bleeding probable cause of cellular death is necrosis, so different biochemical changes will occur. Point is that you can measure time since necrosis/apoptosis started by careful measurement.