T. Foster and Philotomy Jurament have presented the use of the turn sequence from Swords & Spells as part of the OD&D initiative sequence. What I like about that idea is that the structure of a turn reduces some of the all-or-nothing arbitrariness of winning initiative. Things feel more simultaneous, and different sides have more room to react.
I propose a simpler system: Missiles, Move, Melée, Magic. Why? Because it's simple enough to remember and teach while maintaining some of the tactical elements of the S&S system.
Missiles: fire prepared missiles or prepare missiles to fire later.
Move: move up to full move value or prepare missiles to fire later.
Melée: close combat or other actions.
Magic: cast spells if you haven't taken damage this round.
Missile reloading: bows and thrown weapons need not be readied. Light crossbows and slings take one reload action (move or missiles phase). Firearms and heavy crossbows take two move actions to reload.