Why am I getting only 1.5V across the two terminals of the Load even though the Input voltage is 12V?
Q1 and Q3 are emitter followers - this means that the likely voltage coming from the emitter to the load is going to be the Arduino's GPIO logic level minus about 0.6 volts. You won't get 12 volts because those transistors are not configured to work as "switches".

Try PNP transistors and a level translator: -



Here's a typical example of a H bridge using this idea: -



This one drives a motor as load and uses quite powerful transistors but it's scalable to lower power needs. Note that the lower transistors (NPN) are driven directly from logic levels.

As an aside, you appear to have a 6V tapping on your secondary and this will produce a peak voltage of about 8.5 volts and, after rectification the DC voltage will be more like 7V and not 12V.