"d'Erlon at Quatre Bras" Topic

If Napoleon had not ordered d'Erlon and his I Corps to join him at Ligny on June 16th (or if the order to do so had gone astray) and d'Erlon had marched to join Ney at Quatre Bras, about what time would he have arrived on the field? And from what direction would he have arrived?

It's tough to say. He would have arrived from the south as he was marching up the Brussels highway from Gosselies through Frasnes to reach the field of battle.
D'Erlon's corps was late getting moving, having to wait for Reille's corps to depart, with the lead elements (Durutte's division) arriving at Frasnes (in Ney's rear) at approximately 4pm having received those orders to move up to Frasnes from Ney at ~1200.

It was here, at Frasnes at ~1600 that D'Erlon received orders from Napoleon to move towards Sombreffe and fall on the Prussian right. Thus from ~1600 until ~2100, D'Erlon's corps marched and counter-marched between both battlefields, assisting in neither battle. An ADC from Napoleon's staff (identity is under some speculation) had encountered Durutte's division as it approached Frasnes and directed it east to comply with Napoleon's orders, and causing the trailing divisions to follow suit. Had the Corps not been intercepted by that ADC, it would have been assembled in Ney's rear, probably by 1830, at which time Wellington was suitably reinforced and able to take the offensive. Frasnes sits approximately 2.5 km south of Quatre Bras, so Ney would have had a significant numerical advantage by 1930 with which to meet Wellington.

Had Ney not recalled D'Erlon, then his full weight (~20,000 men with 46 guns) would have fallen on the Prussian right/rear, based on the route of march and the relative positions of the armies. The two battelfields are only separated by ~10km so Durutte's division at least could have been in action on the Prussian flank between 1800-1900. It is widely believed that this event cost Napoleon the campaign. The Prussians would have been utterly mauled Had D'Erlon's corps arrived at Ligny and their route of retreat would have been more to the east by necessity, thus increasing the gap between the allied armies. Wellington would have then most certainly fallen back on his lines of communication as he was particularly sensitive to the protection of the Channel ports.

If you look at this map of Ligny, you will see the relative position of D'Erlon's corp on the left. Most likely coming up the Old Roman Road.
Tippleskirch would have been flanked, however Krafft, Bruse or Bose could have moved to the flank to hold him off.

The Prussians would have had to break off the battle earlier, but still moved up the Sombreffe road away from D'Erlon's attack, as they did after the battle.

technically Drouet was asigned to Ney's wing of the Armée du Nord, and Ney planned his battle at Quatre Bras just with the availability of that corps as well.
It is always difficult to say what would have been – but either an active role at 4 Bras or Ligny would have had an huge impact on the Prussians respectivly Wellington's multi national army