I'm a native english speaker, and prompt is pronounced "promt" ( /prÉ’mt/ in my roughly General American accent). Ie, there is a silent "p", but it's the second one, not the first.
In my Ohio dialect (which is supposedly standard), I pronounce both p's, but the second one is only half-pronounced. If I pronounce it quickly, there isn't the slight puff of air between the two, so the sound has the beginning sound as a first bit of 'p' and the ending sound of the last part of 't'. This sounds very similar to /promt/, but is not actually the same.