Perhaps try a premium hdmi cable.
I use Monoprice cables. I did test this last night with my official old school MHL adapter . i used a panny plasma display 2010 year model and did not experience lag as the lag you suggest. i used files that i had used on previous phone (htc evo 4g lte for example) that i just copied over to the ONE.
i use dice player, the stock player and MX player most of the time.
in conclusion, i am simply not seeing the delay you speak about, using the MHL adapter or even wireless using the media link.
since our phones' hardware are identical, perhaps you have files that are encoded that takes a lot of CPU power to decode, thus slowing things down a bit, or you have a slower and/or faulty hdmi cable. all speculation of course...