Weather wars are going to take place between the same Pokemon as before, since Stat Swap doesn't change abilities. It's just that the Pokemon that get the weather starting abilities are now different, and the Pokemon that counter them are also now different.
Sand: Tyranitar and Hippowdown are now both special attackers. Tyranitar has the better bulk of the two and is the fastest Pokemon in Stat Swap Standard with both Stealth Rock and Taunt. Hippowdown has higher speed and Special attack than Tyranitar, but much less physical bulk, and a little less special bulk. Sandslash is the main Pokemon to take advantage of sand support to my knowledge. Cloyster can be run in sand now as it can have Overcoat thanks to its new stats making Skill Link useless.
Sun: Ninetales is now just a little different. the main things to remember are it is now bulkier, has 73 speed, and might be packing physical moves since it has 100 Attack now. Tangrowth is one of the best things to take advantage of sun support with 125 Special Attack and 100 Speed and a good special move pool. Other than that expect a lot of fire types, and whatever someone thinks would be good if its water weakness was covered.
Rain: Politoed is basically the same Pokemon except now offensive variants will be able to be banded, and physical sets may be justifiable. Relicanth is probably going to be banned with Swift Swim since it has 100 Speed and 130 Special Attack Same for Carracosta with 133 Sp Attack and 74 Speed. Look out for Swift Swim Mantine to be banned as well with its 65 speed and 140 Attack. I'm predicting after bans that Mantyke will be one the best rain sweepers with Swift Swim and 120 Attack. Nothing replaces Gyarados, but if you'd like to, you can try Gyarados for that (read that one a few times).
Hail: Abomasnow's offenses take a hit, but hey, at least it has 90 Speed now. Hail is probably going to suck, but there are some Pokemon that would enjoy the support. Cloyster is pretty much the only thing you should be Blizzard spamming with using its 180 Special Attack. Think of Alakazam as the new Reuniclus, but with the kind of move pool you get when you were introduced in Gen 1. Ice Body users are kind of just bad, but Regice has 200 attack, so you can try that if you want.
Countering Weather: When it comes to trapping say good bye to everything you once knew, and say hello to banded Gothitelle. It can one shot Tyranitar (provided it doesn't have dark coverage) with Brick Break. and handles the other starters fine by just bashing them with Zen Headbutt. Lickilicky is faster stronger and built to last longer, with 110 speed and offenses each boosted by a little bit, it can actually do something, and Cloud Nine becomes relevant again, the other Cloud Nine users still suck though, so don't get too excited. Chandelure will be a very solid switch in for Ninetales, it has the kind of bulk that doesn't care about Earthquake which is good since Ninetales only has Dig, and a decent support move pool. Rain Teams should be on the lookout for both Lapras and opposing Gyarados, oh and whatever grass type someone scrounges up. Countering Sand is just a matter of having a decent team as sand overall doesn't seem to be the greatest. Sand used to be limited to mostly physical attackers, now it's limited to mostly special attackers.
