So I do a lot of 'one player per mech' narrative games. What I do:
Each player has their own initiative. So everyone rolls at the start of the turn, and init goes by player. The active player moves all their units when it is their turn in games even when each player has multiple units. (In multiplayer btech games where every player is talking on their side who should move next, or what to save, you waste a lot of time. With individual init, when its a players turn, they go, no discussion or debate, no moving half now half later after Tommy has moved one or two. It works great.)
Some people use cards instead, but having done dice and cards, pulling individual cards and all that just adds more time delay and time spent collecting and passing out cards. With init, everyone rolls dice they already have, no extra card steps, and lowest goes first.
I also usually play as the GM/numbers guy. It helps to have someone who has the common charts memorized. So when people are rolling batches of hit locations, I just look at the numbers and tell the players what the locations are. So if Billy is shooting Tommy, Billy will just say 8, 5, 10, and then ill say to Tommy '5 damage to left torso, right leg, left arm' while Billy gets his next weapon ready. So 3 players may have rolled up hits, but only 1 person is usually calling out locations at once to reduce chaos, with everyone set with their hit locations rolled on the table up waiting for their initiative to call out the locations.
Finally, I run the bad guys like the bad guys in DnD. They have a set routine the players know about, and all the enemy forces go on the same initiative. So when im the GM, each player has their turn, and then I, running the op for, quickly move all the goons and such with their routine. Its 'move towards someone who has lost initiative and already moved, and shoot the lowest number to hit.' Speeds up the decision paralysis part, and lets mechs tank or draw enemies away depending on what the players choose to do. Most importantly, it plays fast, with 12 opfor goons moving quickly versus the 4 player hero mechs.
Other then that, its simple stuff like having movement dice for everyone, multiple pairs of colored dice to batch roll, with little baggies to lend to people matched d6 dice sets, d8/d10/d12 for crit locations instead of upper/lower, roll again. I also tend to watch the heat of the players and tell them when they have overheated, as they often forget to track heat correctly.