The rules aren't particularly clear; thus, it is up to the GM
What the rules do say about command words is as follows (DMG page 141 "Command Word"):
A command word is a word or phrase that must be spoken audibly for the item to operate. A magic item that requires the user to speak a command word can't be activated in the area of any effect that prevents sound, such as the area created by the silence spell.
This is under the "Activating an Item" section which states:
Activating some magic items requires a user to do something in particular, such as holding the item and uttering a command word, reading the item if it is a scroll, or drinking it if it is a potion. The description of each item category or individual item details how an item is activated. [...]
These rules don't really tell us, well, anything about what happens with multiple items having the same command word. As such, it would be up to the GM what happens in this case. Perhaps worth noting is that the section is called "Activating an Item" and not "Activating Items" but that seems like too small a detail to really pay any mind.
Notably, there are also at least somewhat similar things that can already be done, such as setting up multiple glyphs of warding to activate on the same command word. This doesn't really say anything about Magic items, but it certainly makes activating multiple at once less unprecedented.
You could change the action-economy requirements
Theoretically, you could make the items not require an action at all and only require speaking. While this would certainly change the items compared to their appearance in the books, if they're already being custom-crafted or collected it isn't such a drastic change. I mention this as an option because it isn't unheard of in fifth edition; an example of such an item is the Pole of Angling which has been discussed in the following:
Does the Pole of Angling's command word require an action?
For that matter, you can honestly change anything since you're the writer and can take creative liberties. It may not be the most accurate to change the action-economy requirements or to have multiple magic items activate at once, but you, of course, have the freedom to do whatever you'd like.