Hmm, here's a way it could go... sorry for any mistakes:
23 May
Lutjens decides to take the eastern path around Iceland. His decision is predicated on making it past the RN forces to Saint-Nazaire as soon as possible.
24 May
1120: Aircraft flying search out of Iceland spot two large ships proceeding south-west; the contact is radioed in.
1135: Tovey (Home Fleet) confirms the sighting and launches scout aircraft from Victorious. Leaving the carrier with four DDs escort to follow, he turns towards the contact and deploys his cruisers ahead. Holland is signalled and told to make all speed east.
1420: The shadowing aircraft out of Iceland has to return to base.
1455: After a short interval of no observation, the scouts from Victorious arrive over the Kriegsmarine task force.
1510: Victorious' task force has an altercation with a submarine. This is believed to be evidence of close coordination between surface and submarine arms of the Kriegsmarine; it is in fact simple luck.
1520: Cruiser Galatea sights the German task force.
1525: Prinz Eugen alters course to attack the Galatea. The heavy cruiser begins engaging the Royal Navy ship at long range, and the battle is inconclusive for the next half hour or so. One 8" shell hits Galatea, disabling X turret, and another damages her mainmast; Prinz Eugen takes two 6" shells on the belt, to superficial damage, and one shell lands on her upperworks starting a fire.
1600: HMS Aurora and HMS Kenya arrive. Prinz Eugen opens the range once more, returning to station alongside Bismarck. Her final salvo holes Galatea below the waterline with a non-fuzing shell, causing Galatea to slow for repairs. Galatea takes a course north and west, away from the German ships and towards Iceland.
1610: A second wave of aircraft arrive from Victorious; these are armed with torpedoes. Of the ten Swordfish launched, two are damaged and forced to drop early and one is shot down. Most of the remaining torpedoes miss, but one holes Prinz Eugen near the stern and shock damage renders her right hand shaft inoperative. (This means Prinz Eugen is reduced to her central shaft, and she slows considerably.)
1645: Lutjens elects to order Prinz Eugen back to Germany, and continue along with his flagship alone. This is partly due to the fire burning on the heavy cruiser, which will draw attention at considerable distance; if he can reach the French Atlantic coast, he will have largely succeeded in his mission to break out, while Prinz Eugen manifestly cannot.
1705: HMS King George V opens fire.
The engagement is not particularly complex. KGV leads Repulse, and begins by staying inside her broad immune zone against the weapons of the Bismarck.
Both battleships straddle quickly, the KGV with her third salvo and the Bismarck with her nearly contemporaneous second.
The German BB scores the first hit, a shell hitting the KGV's belt but failing to penetrate. KGV's reply with her seventh salvo scores a hit on the Bismarck's bow, passing through without fuzing, and her eighth strikes the superstructure.
Bismarck does the next damage, hitting A turret on the KGV. This jams the elevating gear on one gun, but the remainder remain operational.
Bismarck is slowly turning to port as the battle continues, cutting inside the British capital ships and aiming to turn away to the south. To this end, Lutjens splits his fire now that Repulse is in range - the battlecruiser is faster than his ship, and must be lamed if he is to escape.
Unfortunately, however, his first hit on Repulse does not do as much damage as could be hoped - unlike her fellow battlecruiser the Hood, Repulse has been regularly refitted and has extremely thick armour over the magazines, and the shell which scores a lucky hit near the magazine fails to penetrate four inches of high-tensile steel.
Shortly after this, Repulse scores with her 15" guns on Bismarck's Y turret, disabling the turret.
Bismarck fights well, but the outcome was never especially in doubt - with sixteen guns to eight, and against better protected ships, Bismarck's only hope was to disable the battlecruiser and escape.
Guns silenced, battered into a wreck, she is sunk by torpedoes at 1839.
The two British capital ships do not escape unscathed.
KGV's battery is significantly damaged by the engagement, being down to five operational guns (though two more are only somewhat incapacitated and function is restored before she makes port), and her loading speed draws criticism towards the end of the engagement. However, only two shells penetrated her belt, one where a previous shell had already damaged it, and no shells successfully plunge into her vitals.
Repulse has taken several hits as well. One of these hits is a plunging shell under the belt, which damaged the machinery - had it been scored earlier in the engagement, Bismarck might have been able to break off and escape. B turret was also disabled by shellfire, and she has lost her mainmast - and her aviation facilities, which result in a nasty fire.
Tovey comes under some criticism for his handling of the battle. Some feel he should have waited until a second air strike went in (or until Holland caught up), or that the first strike should have been directed at the Bismarck. He is also accused of being glory-hungry for sending his flagship in first.
This is all behind closed doors, however - publicly, he is hailed as a hero for having saved thousands of British sailors in the convoy routes (and for sinking the largest battleship yet sunk at sea.)
Goebbels makes much of the successful return of the Prinz Eugen to Norway; however, it rings somewhat hollow as the Bismarck has clearly been lost.
King George V is sent for repair in New York Naval Yards, for maximum propaganda value; Repulse, shorter on fuel, returns home for her repairs.
Once Repulse is out of the yards, Hood gets a much needed refit - it has not escaped notice that the hit Repulse took early in the engagement would have blown Hood to pieces. She is in refit until early 1942.
Prince of Wales finishes her working up and sea trials, is sent east, and is unfortunately lost returning to Singapore after the Battle of Eastern Malay in late 1941. (HMS Renown makes it back to Singapore after the battle, but is dry-docked for months having torpedo damage patched up.)
