While it is true that if the Federal reserve bank makes a change in their rate there is not an immediate change in the other rates that impact consumers; there is some linkage between the federal rate, and the costs of banks and other lenders regarding borrowing money.
Of course the cost of borrowing money does impact the costs for businesses looking to expand, which does impact their ability to hire more workers and expand capacity. A change in business expansion does impact employment and unemployment...
Then changes in employment can cause a change in raises, which can cause changes in prices which is inflation...
Plus the lenders that lend to business see the flow of new loans change as the employment outlook change.
If the costs of doing business for the bank changes or the flow of loans change, they do adjust the rates they pay depositors and the rates they charge borrowers...
How long it will take to change the cost of an auto loan? No way to tell.
Keep in mind that in complex systems, change can be delayed, and won't move in lock step. For example the price of gas\s doesn't always move the same way a price of a barrel of oil does.