How do you finish a stalemating game?
Solution 1:
Sadly, there's no sure-fire way of resolving this situation.
You won't be able to rely on minions to take out their towers and nexus, especially if their uber status gets to the point where they can start defending their respawning inhibitors. If you get really lucky and get their nexus towers down, you could just farm minions hard enough for yours to take the nexus down a bit then try to dive the nexus. I'd consider this a Hail Mary more than anything else, though.
If you know you're going up against a team that is going to steamroll you once they get maxxed out (2+ carries), the main goal is to take advantage of the fact that they will be weaker in the early game. If you zone the carries and/or kill them enough to keep them from getting geared and leveled, your team will get powerful enough to take their nexus before they can do anything about it.
Assuming you find yourself in that unfortunate position anyway and forcing their team to split up isn't working, you absolutely have to focus the primary damage dealers down first. If they stick together and you're not killing their carries first, you'll take way more damage than you will deal out over the course of the fight and you're sure to lose.
Solution 2:
Once you have reached this point, the team with the better comp is usually going to come out victorious, but like you said the trick is to try to just catch one of them off gaurd and kill them then just push, at 3v2 no matter what their team builup is they will lose at this point in the game. The trick to sway the game for you though is to play very conservativley and ensure noone on YOUR team gets killed, and also don't hesitate to change up your build. Just because A build works 90% of the time, maybe you are in a situation where a build change will be whats puts you over them. You will have a ton of gold by this point and will be able to buy pretty much anything you could need. For example if you usually play a tank and build a nice well balanced build and you are up against two heavy hitting ranged AD's then just sell any MR you have and put it all into armor. LoL is dynamic and should be played as such. Never tie yourself to a build, you should always be trying to counter the enemy. No team comp is unbeatable. Prioritize enenmies, change your build and outsmart them.