Last month, on Oct 23, I connected a fresh LR44 button cell up to one of my two transistor flashers. It ran up until Nov 18 for a total of 26 days. That’s about 624 hours. Assuming that the LR44 has about 150 maH capacity, then the flasher is drawing about a quarter of a milliamp. But I can’t say for certain what the average current is because as the cell voltage drops, the current also decreases. It may start out higher than 1/4 mA and end at less than 1/4 mA.
I used a clothespin for the clamp to hold the wires to the button cell. I drilled small holes through the wooden ends, and stripped the wire, put it through the hole and bent it over so it made contact with the cell. Previously I used screws and nuts, but this does an adequate job and is a lot simpler. You could use any plastic clip with enough width between its jaws to hold the cell and wires. It just has to be an insulator, such as wood or plastic. A paper clip could be used it it is wrapped with some tape to prevent the metallic wires from shorting the cell.