In the other games, pushing a focus power check always increases the severity of the associated psychic phenomenon. I can't seem to find such a thing in Deathwatch.

It doesn't, you can find the rules on pg 185. The character gets an automatic psychic phenomena, in addition, if they roll doubles they take a level of fatigue.

If you're looking at how to convert Psychic Powers in DH to DW, you may wish to consider reading the Rites of Battle P71. It doesn't supercede DH, just provides and alternative to porting over to DW.