Electronic – Battery Charging Time Computation

You do need more than 12 volts to charge a 12 battery. Also how do you know that your solar panel is producing even 12 volts? Just because that is it's rating doesn't mean it always produces that value. Do you have enough sunlight? Have you measured the output of the panel? Can you provide more information on the charge controller? If you did provide 24 volts to the charge controller, how much voltage will it apply to the battery which is important because 24 volts is probably too much. Your panel is rated at 12 volts and 5 watts. Thus, assuming it is getting sufficient sunlight, its maximum current is 0.42 amperes. At that level, charging a 4.5 AH battery, would take a minimum of almost 11 hours.

You need to add more detail but I will say that this will probably not work. I'm going to assume that your batteries are 12V lead-acid batteries, please edit your question if that is not correct.

To properly charge a 12V lead-acid battery you need a charging voltage of at least 13.8V and something like 14.4V is better. So, you would need a charging voltage of about 28V to 30V. If your 12V-24V converter really supplies 24V then it will not be able to charge the batteries.

It's worth mentioning that you need to be careful when charging batteries in series like this. A fault in one of the cells can cause all of the other cells to be overcharged and destroyed.