I'm not really certain what you mean by do they work with the relaunch.
I only read the original four novella series when they came out about thirteen years ago. I remember there was something a little different about them, though I couldn't put my finger on exactly what that was and it wasn't a bad thing. I really enjoyed them. I have recently picked up the entire series as ebooks, including all the mini-series and cross-overs. I'm not really certain why I've waited so long and am really anxious to get started. I'll probably start with the original four again and just read straight through. It should get me well into next year!
As far as fitting in with the other series, yes, I think they do. The sort of had their own continuous on-going story. Over the years there've even been cross-overs between New Frontier and other Trek series and other series have mentioned characters and events from the New Frontier series. There was even a couple of New Frontier novellas in the last two Mirror Universe anthologies.
Having not read past the initial start-up, I can't say for certain what all happens but if memory serves me, I believe the original Excalibur was an Ambassador class, it gets destroyed at some point in the series and then the crew gets a newer, suped up Galaxy-class.
Pick them up and give them a try. I don't think you'll be disappointed.
I hope this helps some.
- Byron