[RPG] How does time pass in combat

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.

Tools to add to your DM kit to keep combat moving

I suggest that you add three DM techniques to what you are already working with:

I try to always give a heads-up to the next player down the initiative order as the current player's turn starts

While that's nice of you to do that, and you can keep doing it (no need to stop), the players need to take responsibilities for their own actions and their own attentiveness. You have an entire opposed battle to run, they have one character.

• Tool Number One: Increase Player Initiative Awareness

  Enhance the visual representation of initiative so that it's right in front of them with their name on it. See this answer for a number of awesome techniques to keep initiative accessible. These tools help.
  The passive initiative Quadratic Wizard suggested I've seen work at the table, but it depends on what skills and feats your PCs have, and if they are content to always be in the same turn order.

  You can have the players change seats with each initiative roll, so that you always go clockwise, or counter clockwise around the table to call on them, but how that is received is very table dependent. I have only seen this done successfully once; the other times players rolled their eyes and crack jokes about playing musical chairs or D&D.

In large combats, treat groups of weaker creatures as mobs ("Ok, all 4 of them attack Bob" and roll attacks and damage together)

While that can lead to bursty damage that may create other problems at low levels, there's no harm in doing this. This will help to speed up combat.

Encouraging players to make a decision if they are taking a long time to decide what to do.

You are smart to do this, but I'd take it a step further: don't encourage, require a decision. At some point, you tell the player to

• "Make a decision; you have 15¹ seconds or you take the Dodge action and it's the next player's turn."

  I'll elaborate on this further down. This is the single most effective way I've seen to both speed up combat and increase the sense of urgency and danger.

I produce cards for all the PC's active abilities and spells, to make it easier to look up what they can do

That's awesome of you as DM to do that. Tools like this help to avoid delays. You can call on your players to create a handy reference tool for their character. (Player motivation on that varies greatly) Aside: I got a few of the 5e spell card packs from {that online river merchant} and they are very handy, particularly for new players.

Two More Tools that should help speed up combat.

• Tool Number Two: Enforce a time limit, but allow two² questions before a decision.

  • What's the point of the questions? The players need to make informed decisions, and you are their source of in-world information beyond the characters' declaring actions of telling each other what their are doing/have done. "How far is that ogre" or "Can I see the priest over there?" or "Do I see any other creatures next to that beholder?" are the kinds of information that the player needs to decide what to do on their turn. So too is a question like "Do you need help with that roper or not?" to an adjacent player.
    
    

  Applying this tool requires a new "session zero" before your next play session. The information that you are passing to your players is as follows:

  • You have to pay attention to combat; it comes at you fast and furious. A lot can happen in six seconds.
  • When it's your turn, be ready to tell me what you are doing. You get to ask two questions{2} before declaring an action. You can ask me, or another player, either of the questions.
  • If you don't make a decision within a brief time¹ (10 sec, 15 sec, 30 sec) after the questions are answered, you take the Dodge action and it is the next player's turn.
  • There is no reason to ever miss your turn. If you pay attention to what's going on, and already have in mind an idea of what you are going to do, most of the time you won't have questions and when you do they will fit into the situation, and the decisions you are already thinking about making.
    
    I've seen this work in 5e as a player. I've used it as a DM in multiple editions. I learned it as a player from other DM's. When the DM began to count down "5, 4, 3, 2, 1..." decisions got made. Some great ones, some awful ones, and some memorable ones. (You threw the burning hay bale where?)

• Tool Number Three: enlist the aid of a player to track things.

  Whenever we have done this, I've seen it speed up combat, particularly as groups got bigger. The trick to this is in asking a motivated player to help track things. Track initiative; track HP; call out the roll's result. Picking the player to do this is probably the hard part, as some players do not want to do this and they should not be forced to.

• Tool Number Four (optional) that maybe helps: dice cups.

  Quadratic Wizard suggested rolling ahead of time, but this is slightly different than that - I've done this once successfully with new players who were junior high aged. (Different edition, same funny shaped dice).
  _{The use of dice cups I had learned playing drinking games / bar games with dice in the Navy.}

  • The dice a player rolls are always in the cup. As you suggest, warn the player that their turn is next.
  • Player shakes up the dice in the cup vigorously, with one hand covering the opening.
  • The player then more or less slams the cup down on the table leaving the dice still covered by the cup.
  • The player then declares what they are doing, where, and to whom.
    
    This allows for quite a bit of description. (In this particular group it encouraged the narrative elements of role play).
  • The player then lifts the dice cup. The DM reads the dice, and then narrates the result.

• Since I didn't use this in the current edition, this tool may or may not fit your table. In our case, it (1) helped to keep play moving, and (2) got the players to narrate what they are doing before we saw the dice results. Worth a try, but I can't guarantee it will work with your group.

¹ Set the time limit to 15, 20, 30 seconds, or whatever (it helps to get player buy in on the time limit). Then stick to it.

² You can allow 3 questions, or 1 question, but 2 is the number I've seen fit this best.