I'll preface this first by saying that electromedical devices is not my specialist area, other more experienced Covers can correct me if I go offtrack at any point.
The global requirement is to demonstrate compliance to IEC601-x / IEC60601-x.
The "UL Mark" is pretty much set in the public conscious as evidence of compliance. I have no idea of the history here, but clearly there was eventually an issue with a commercial company having a virtual monopoly over the provision of a testing service.
So other organisations can also offer testing to those standards, and I believe that they can also offer the UL mark.
So you should definitely shop around for the testing organisation that best meets your requirements.
But the requirement to demonstrate electromedical safety is of course different to the additional requirement to have an ISO13485 (and/or FDAQSR) compliant quality system in place. After all a DNA sequencer can be perfectly safe, whilst failing to actually sequence correctly!
So as the electromedical testing will be additional in any event, the bottom line is not to get solely fixated on achieving evidence of electromedical safety, it is only part of what you will need to do.
Thus your ISO13485-compliant system should include the plans and processes required to create and maintain a design control system that includes achieving evidence of electromedical safety, as one part of the overall design programme.
I would advise starting at the very top, flow-mapping a strategic overview of the key processes within the organisation, and then cascading flowmaps downwards as things start to take shape within the organisation.
You should be able to map your critical design control flowpath at a relatively early stage, to get you off to the best possible start.
Of course if the machine is already built, (i.e. without having design control procedures in place) it is possible you will have some sizeable holes in your quality systems evidence. You should definitely create the procedures anyway, and do a Gap Analysis to ensure that the design programme (as was) can be justified as having been adequate. If it can't, then you will be able to do some corrective action to patch any holes.
Obviously any design work that is going on at the moment, especially design validation and design transfer operations, should certainly be done against robust procedures .
Design transfer would include the various quality plans required to manufacture the device on a routine basis.