They need a Reaction unless they close their eyes earlier
The DM may rule that your team has trained this enough so they can do it without needing a Reaction. Our DM did allow it. I just think you cannot point to rules as written for demanding it.
However there is nothing that stops the other party members from closing their eyes. Closing your eyes is so minor that it would be a free action, see PHB, p. 190:
Your turn can include a variety of flourishes that require neither your action nor your move. You can communicate however you are able, through
brief utterances and gestures, as you take your turn.
The DM might require you to use an action for any of these activities when it needs special care or when it presents an unusual obstacle.
Closing your eyes certainly is a lot less involved than talking. A malicious DM might argue that if you do not take special care to keep your eyes closed in the fracas of fighting, then you might glimpse the pattern, so it needs an Action. I think that would be stretching it.
If other characters state that they close their eyes at the end of their turn (and those after you in the initiative order do so at the end of their turn in the round prior), then no Reaction would be required to do so, and they could open their eyes again at the start of their next turn, not losing any Actions.
They would however be blinded until they can open the eyes again on their next turn, and suffer the appropriate disadvantages.
I think it would be normal by RAW to demand a Reaction, if characters want to do anything that involves risk outside of their turn. So if they wanted to wait for you to finish your casting and then open their eyes, or wait until you start and then briefly close them while the illusion flashes up, the DM would be in his rights to require a Readied Action for that. This of course would preclude them from doing too much other useful things that turn.
So: either they suffer the blinded condition, or they forgo their Action for a Reaction.