The second scenario is more or less OTL - as soon as the Grand Fleet showed up, the Germans started trying to get away.
So let's go with the first scenario. Jellicoe is delayed somehow - let's say a submarine scare, they don't go home but spend an hour milling fruitlessly around the North Sea trying to avoid a submarine. The existence or otherwise of the submarine is irrelevant, they saw plenty of imaginary periscopes IOTL.
So, you still get the Run to the South and the Run to the North as OTL, including their losses. Beatty turns to the northwest rather than the northeast to fall back on the closing Grand Fleet, with Hipper in hot pursuit and Scheer trying his best to keep up. Given the changed geometry from OTL, it's likely that Jellicoe deploys to starboard, Hipper recognises the peril into which the HSF is being led, and the German forces make their best effort to leave nineteen battleship-shaped clouds of clinker over the Jutland Bank and contrails on their way back to the Jade.
Perception of the battle in Britain is that Beatty did his job but was let down by Jellicoe showing up late. The Germans will claim a win even more vociferously - British losses are probably two battlecruisers to nil for Germany, unless Hipper lingers too long under the guns of the Grand Fleet. The Royal Navy will have learnt its' lesson about powder handling on the battlecruisers, though, and Der Tag has not yet come. Both sides will probably keep trying to set up a fleet action, quite possibly fruitlessly.