I'd have thought with just a single application (not involving dependants), you shouldn't have too much trouble getting the date you want, provided you book as soon as slots are released. Try Croydon PEO first, as it's biggest and local to you. Try logging on a few days before 6 weeks, as they sometimes release a bit earlier. Usually they are released overnight on the relevant dates, so stay up late!
UK partner's absences have no bearing on naturalization, unless there are reasons to doubt the durability of relationship - i.e. they are in fact separating.
You have to get a long-term residency in Ireland, and to be eligible for naturalization, it has to be a category such as employment, self-employment or marriage, not student or visitor. And you have to explore ways to find sponsorship for work permit, for which you need a specialized skill and experience in short supply, or qualify under a shortage occupation.
Regards,
Delfina