I've not done a half marathon but I did do a half marathon length swim (5km, took me about 2 hours).
I kept my training really simple. I started 3 months before, with a solid base swimming skill and the benefit of being young but without huge baseline fitness (office job, sedentary hobbies). For reference on fitness levels I did a 2.3km swim the previous summer with no training but was so wrecked I could barely haul myself out the water by the end of it and ended up needing glucose gels 3/4 of the way through. Whereas this swim I did on a big breakfast, electrolytes and a massive dinner after, possibly one gel.
I started at 500m and just added 500m per week to my long swim. Swam 3x a week with my short swims starting off the same length as my long one (500m) and ending up at 1-2km (equivalent to 4-8km ish) by the end.
Stopped with the long swims 2 weeks before at 4.5km.
I didn't do any interval training or any of the stuff that it was recommending on the training plans online because I knew I didn't enjoy the sorts of swimming they were recommending and I knew the only way I would train was if I enjoyed it, at least a moderate amount.
I don't know if that helps given it's not running, but I guess I concluded the most important thing was getting out there and building my stamina up, not really which way I built it up.