John - unless your institution is a 'power-house' of it's own research and publications - then publishing companies are unlikely to assist you setting up a journal. More conventionally, is that institutions plan (and it takes several years' to plan and create through a 'community of existing internal editors) their own 'in-house' university press journal. It's where the likes of Cambridge Univeristy Press and Oxford University Press came from - but they are rare examples.
You can form a partnership with a local (non-research) publisher to do the printing (at a price). Use your contacts to do the English checking and refereeing. The difficulty is in getting enough material for Volume 1 Issue 2 of the "Male Nurse Researcher" or whatever you want to specialise in. You can use your research students to submit papers, and reward them with points towards their degrees.