I have just learnt about the demand and supply framework for money and how the equilibrium interest rate in an economy can be determined. In theory, based on that framework, an increase in nominal incomes would lead to an increase in the demand for money, causing interest rates to rise. However, in reality, I don't see that occurring. In fact, in reality, say in the United States, rates are often set by the Federal Reserve through manipulation of the money supply. To me, rates don't seem to change as a result of changes in nominal incomes. Why is this so?
If you are referring to wages when you say 'nominal incomes' then you are reversing cause and effect. What happens first is the intervention by the central bank to artificially lower interest rates which ultimately increases the money supply. When the extra liquidity finds itself in consumer goods, the cost of living increases for the average wage-earner. It's at this point that labor requires a new equilibrium price and nominal wages increase. This can take months or years to play out but traditionally wages are always the last set of prices to rise.
Models based on supply and demand little insight into real-world interest rate determination. The most important factor (but not only) in determining the yield of bond that is free of default risk is rate expectations: what is the expected cost of financing the bond at the short rate over the lifetime of the bond? Once that factor is accounted for, we can then turn to other considerations will influence the price. These other factors are lumped into what is called the “term premium.”
This is a disputed topic, and some academics will make arguments to the effect that “rate expectations do not determine bond yields.” However, that is largely equivalent to saying that the term premium is not zero, which is not really disputed. You would need to search through the literature on the term premium to get an idea of the magnitude of term premium estimates. However, from the perspective of people who are not familiar with bond markets, most term premium estimates are small (less than 1%), and so the thing they are most interested in are the level of rate expectations.