There are 2 definitions of "round" in 5e. There is the definition of "round" that you quote, which is from initiative position intMax to initiative position 0 (or negative if you manage that some how).
Then there is the definition used here for readied actions. This is the same definition used in "once per round" effects such as certain powers. This definition begins at the beginning of your turn and ends at the beginning of your next turn.
Because (unlike in 4e), readying an action and activating it, doesn't move your initiative order position, and because it matches the readied action refresh timer (which happens to use the same 1/round definition of round), readied actions can be used at any point in either the current round or the next round prior to your turn.
Here's how reactions refresh:
When you take a reaction, you can’t take another one until the start of your next turn. If the reaction interrupts another creature’s turn, that creature can continue its turn right after the reaction. (Player's Basic p70)
This is a large part of the support that leads me (and many others) to believe that readied actions can roll up to your next turn.
Readied actions happen when the trigger occurs, interrupting the initiative order. After the readied action takes place the rest of the action takes place as normal.
In the case you mention, it is valid for one player to move in response to an enemy attack to give cover, given that he/she had readied the action (or has an ability that allows him to use a reaction in the same way).
Although I think he would efectively be giving cover to his ally, not taking the brunt of the hit, because the enemy would have been aiming at the ally, not the reacting player. It doesn't make complete sense, but it is that way according to the rules (they are in place to make thing easier and simpler, with as much realism as needed without being bothersome).
Your reacting player would give a +2 to AC to his friend. Then, by optional rules, if the attack doesn't hit the protected ally, but would have hit him if he hadn't the cover (the +2) it hits the cover. In that it would hit the reacting player, but it needs to surpass his AC to damage him.
It is a bit convoluted, so tell me if I need to clean up or clarify my answer.
And, of course, you are free as DM to change anything and allow the reacting player to completely protect his ally despite any rule I have commented ;)
Best Answer
RAW: Unclear
The rules for this are in the PHB, which mostly deals with PCs, which in turn usually can’t get multiple reactions (the only ways which can grant you multiple reactions tend to be limited to attacks of opportunity, and even those were only written after the PHB was released).
As such there is no language explicitly limiting the trigger to only be able to occur once in a round, even though I suspect that it is the intended reading. I find saying “the rules don’t explicitly say you can’t, so you can” not to be a useful guiding principle so I’d say the RAW isn’t clear on this. I personally would rule that no, you can only use your reaction to react to a trigger you set with Ready once because action economy wise that just makes a lot more sense, but I do not believe that a strict RAW reading of the section yields this as the answer.
RAW probably doesn’t matter in this case
If you decide your Marilith is picking up a longbow and shooting people with it you’re already moving beyond a literalist RAW reading, as of course shooting things with a longbow is not listed in the creature's statblock. This is not a big variation and perfectly in your purview of things to do as a DM, but you have to decide on what the consequences here are. For example, does it still get a Multiattack? If so, how many? Maybe it only gets 3 shots because it needs two arms for each bow? Similarly, the multi reaction thing was very probably themed after it using its many arms with one handed weapons; Does that still even apply if it uses bows instead? Should it still work even if it is using only one bow?
At this point what you need to realise that you are effectively homebrewing a slightly different creature with a slightly different statblock, and it’s on you as a DM to figure out how it should work. There isn’t that much helpful RAW guidance for you here. NPCs aren’t bound by quite the same rules as PCs to begin with, you should just make sure that whatever you do it makes sense in fiction so your players can know what to expect from your description of events.