Every fight begins with a fast turn. After this turn, everybody takes a slow turn, and so on.
In a fast turn, you can move half your speed (round down), make half you attacks (round down), etc.
In a slow turn, you can do the same, but round up. Most actions happen in the slow turn, but, like attacks, they are subjected to "halving" if possible (if you choose to "Dash", for example, you just double your movement for both turns).
It is very likely that you have to declare your action during or after the fast turn.
Let us say a fighter has three attacks and moves 35 feet in a turn: he could move 15 feet and attack once in a fast turn, and move 20 feet and attack twice in a slow turn.
Explanation:
As I've said,the problem is not easy to solve - but easy enough to reduce by half.
For example, instead of of having a whole turn where you get up, move 15 feet, and attack you opponent three times while he DOES NOTHING, a more "organic" solution would be something like: you move 10 feet, he moves 5 feet, you attack twice, he attacks once, you attack once more, then he attacks, etc.
The inspiration for this idea is Shadow of the Demon Lord, BTW.
Weapon speed
Two guys are facing each other, 30 feet apart. One has a dagger, the other a halberd. Who hits first? Obviously, the halberd (unless you THROW the dagger, which is an interesting distinction); but most RPGs that get to this level of detail say that the dagger attacks first (this includes the 5e DMG optional rules).
Which is why I'd say weapon speed shouldn't be about who attacks FIRST, but who attacks AGAIN first. The guy with the halberd might hit first, but the guy with a dagger could stab a foe three times during one halberd swing.
Easiest way to do this: a critical hit gives you a CHANCE to attack again. The faster the weapon (an the greater the number of attack you have), the greater the chance. Or, to make things faster, the GM could roll a single dice every round: every weapon with a speed above that threshold gains an extra attack.
If you want to account for weapon reach, being approached by a foe with a shorter weapon allows you to attack first if you're not engaged with anyone else (an opportunity attack would be an appropriate alternative to this).
Spells and spell interruption
I like the idea of spell interruption, which is not an important thing in D&D 5e. In this system, you could probably choose a spell to use in your fast turn, and only "fire" it in a slow turn. Alternatively, you could only use spells of a (spell slot) level that is half your maximum in your fast turns. For example, if you can cast 9th level spells, you could cast a 4th level spell in your fast turn, OR prepare a 9th level spell to cast in your slow turn.
I'm tempted to say that, in the first case, you could ALSO cast ANOTHER 4th level spell in your slow turn - provided you didn't lose concentration between turns.
Bonus actions/reactions/object interactions
I would probably allow reaction in BOTH rounds. Seems to me that it would be a welcome addition. Same goes for object interactions.
Bonus actions are trickier; limiting to one per round seems more reasonable, but see below.
Choices
One way to make this system more tactical is allowing some choice. Usually, you can move AND attack, but if you decide not to move, maybe you can take an additional bonus action (not more than one bonus action per turn), cast an additional spell (4th+4th, as seem below), or make an attack in your fast turn, if you have a single attack. If you have multiple attacks, you could use a crossbow multiple times without moving, or a net and trident, etc.
Pros
- Shorter turns means players pay more attention.
- The "ready" action makes a bit more sense.
- Tactical choices.
- Organic movement./attacks.
- Spell interruption.
- Interesting use for low-level spells at high levels.
- Action Surge looks better.
- TWF could be "fixed" by something like this (more object interactions, more bonus actions, etc). Same goes for crossbow, net, etc.
- More straightforward than checking the DMG table.
Cons
Frankly, this level of detail isn't appreciated by many, and I'm not sure I'm willing to put up with it. "old school" initiative, with phases and all, still sound a bit simpler to me. Still, for a combat-oriented game, I feel this could be fun.
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