I can't speak for the first one since I didn't play the Gridania storyline, but as for the others, they basically didn't go anywhere since the 7th Umbral Era storyline (GC quests) took over until the end of the game.

I want to say Reinheart translated a post about someone asking about the Ascians (these were also mentioned in the LL storyline) and the rep said they will explain about them in ARR.

Leviathan will obviously be in ARR due to the concept art floating around, but I don't know if the story involving Leviathan will be the same as was intended for 1.0. There was always whisperings about him from NPCs in the LL story, but nothing much ever came of it.

It seems there was a huge disconnect in the city-based main storylines just due to the incompleteness of the game. It seems like the platform for the main story was to work somewhat like how it did in FFXI where you did some missions for your home nation, learned a bit what was going on there, then the attention turned to the beastmen and the Shadow Lord.

Once you defeated the Shadow Lord, the rank missions continued with the major storyline going on with your home nation. It feels like this was what they were going for in FFXIV, but it never came into fruition. In the case of the LL storyline, there was some political things going on in the background with one person trying to overthrow the Admiral, but there was also the threat of the Sahagin. (of whom we never even saw a glance of) Before things could go any further with that storyline, you were ushered off to Ul'Dah to take place in the Path of the Twelve story. (kind of like how you were eventually sent to Jeuno in FFXI to deal with the beastmen/Shadow Lord threat)

I don't necessarily wish for them to 100% copy the stories intended for 1.0, (I wouldn't mind if they did though) but I do hope they touch on some of the major plot points that they left unfinished, such as Leviathan and the Sahagin in the case of the LL storyline, and also the Ascians, which were briefly mentioned in the LL storyline and was seen in the Path of the Twelve storyline.