It wouldn't be a language so much as a giant ass board full of buttons. A fusion between a pilots cockpit and a keyboard, with commands reduced down to button presses and a specifically designed script language.
Firstly, I want to point out that humans are terrible at multi tasking. We don't actually multitasks. We just switch between the two tasks fast enough that it looks like we are multitasking. A simple example if you want to try, read the two sentences below at the same time.
The Fox jumped over the lazy Dog
Betty brought a bitter batter
You are either switching between the two sentences then combining them back, or you read one, then the next. Another example would be to perform complex mathematical equations while reciting the composition of titin (if you can do this than kudos). Or maybe try and play a Playstation Game and a PC game at the same time. While it is possible to perform simple tasks at the same time, these are often muscle memory and highly repetitive (e.g. pat your head and rub your stomach at the same time).
The communication channel should be limited to a single channel to ensure that there are no erroneous readings or accidental triggers. This way, your human operator can devote 100% of their energy and time, hopefully entering into a flow or trance like state.
Secondly, typing is going to be faster than speaking or moving. You have 10 fingers and essentially 2 feet which you will have accurate control over (10 toes if you want to really stretch it). You only have 1 mouth so while you sometimes beat a persons typing speed, you won't be able to beat a specialized script which has been compacted for maximum efficiency (e.g. instead of Move, the command is M) along with a layout which reduce finger movement ( Like a Dvorak keyboard vs QWERTY ). You also remove the possibility of getting tongue tied and your super important commander can also eat and drink while operating the controls, assuming you have the additional machinery required to bring the food and drinks to mouth level.
This is also the issue with tracking body movements. Movement is energy. A person is only going to be able to wave their hand so many times before it gets tired. We reduce the amount of energy required to move by focusing all movement into our fingers, and the repetition will strengthen them over time.
Higher Information density is great, when you need to process information. But when you need to output a large amount of information, you need an easy, reliable, quick and efficient way to do so. Excessive movements from multiple sources creates a huge burden on someones mind and introduces multiple sources of errors. Given humans ability of focus on a single thing at a time, it makes more sense to limit their options.
In that sense, you are only as fast as your slowest command. Given that a fast typing speed is roughly 75 words per minute, with an average of 4.5 characters per word, a person could roughly type 300+ characters. Depending on how condensed your script is, you could likely condense most of your commands into 2-3 characters per command and have an average of 100-150 commands per minute. Anything slower should be cut out as you would be negatively your CPM (Commands per minute) due to the time taken to switch your train of thought and then perform the slower action.
The final result? Your command room consists of multiple monitors surrounding a comfy chair with multiple foot pedals and a giant keyboard with a hundreds of keys, switches and buttons.