I've played hundreds of hours across the full spectrum of Call of Duty zombies, and can relate to various communication problems. I've encountered many problems... playing with someone without a mic, having intermittent mic problems, Xbox Live party crashes... etc. I want to cover these types of games specifically:
There is a fair amount of pre-planning that can be done on any given map. You will need experience in the maps to have an effective plan. You will also need an end-goal, whether that is trying to replicate in-game glitches, getting to high rounds or attempting to perform easter-egg or special weapon steps.
You can also assign (or assume, based on regular playing habits) the areas of the map each player specialises in (such as always sticking to specific windows in the starting area).
You should have at least one person who controls the wave. Generally that is the best player and/or host. You can have multiple people in this role (2 in a team of 4 works well). Their job is to keep the last few zombies alive at the end of each wave by building a train. Basically, you need to run in a particular pattern (often a circular path) to have the zombies herd together and follow you around the map. There are plenty of youtube guides and demonstrations.
The reason for suggesting the host to train zombies is that they will have the best connection. A temporary drop in connection can lead other players to being quickly downed when running zombies in close proximity.
If 1 or 2 are trying to train zombies, the other players can concentrate on killing only. They can help revive each other etc. You don't need a huge zombie train (feel free to pick off members, which can be done by the guy training or any other player). Once you have a manageable number you can then setup for the next wave easily.
You can use the breaks to communicate, restock, toilet breaks, perform easter egg steps and so on. You could keep a single zombie alive, but there is a risk that the zombie will die out and start the new wave when everyone is not ready. The rules really depend on the actual Call of Duty game you are playing (as some of the later ones appear to just kill off the last zombies if you take too long, or have received some damage).
Note, the guy training zombies can let others babysit the train too.
I would imagine that players with communication problems (as well as the weaker players in a group) are better suited to just killing. Allow the better players take on the role of training.
Another consideration is that with players dedicated to training zombies, it really spreads out the horde. There will be less of a burden on players who would otherwise struggle.
On a site note, you can use excessive jumping and/or crouching to communicate. Whether that is to get someone's attention or as a sign that you need assistance. I've used this in the past to let the other guys know I am having mic issues or to get them to take over babysitting zombies so I can hit the mystery box.