Already did it as my first 5E campaign. Started at level 3 and ended at level 18, but it could be modified to fit 1-15 fairly well.
Start at the Villiage of Hommlet, leading into the Temple of Elemental Evil. I would use a very restricted version of Zuggotmoy, since she's imprisoned.
Next, be asked by Veluna to investigate the Lost Caverns of Tsojcanth, before a rival party from Bissel can do so (no one knows what they're looking for, but if Bissel wants it, Veluna wants it first). Add in a side trek to the Forgotten Temple of Tharizdun, then recover Daoud's Wondrous Lanthorn.
Keoland had asked Bissel to recover the Lanthorn, because a divination revealed that it was necessary. Allied Sterich has fallen under siege by giants, and before an organized defense could be raised, the capital city of Istivin has fallen into unfathomable shadow. The Lanthorn is supposed to be able to banish the darkness, and either the party agrees to help because the rival group convinces them, or Veluna sends them as part of a diplomatic agreement (that greatly benefits Veluna).
While an archmage is going to use the Lanthorn to free Istivin, the party goes to fight the giants, as per Against the Giants. The events within lead to the knowledge of the drow interference, which leads Into the Depths of the Earth and the Vault of the Drow. Lolth is revealed to have been the source of the attacks in Sterich, as well as partly responsible for the Temple of Elemental Evil.
In the Demonweb pits, they find that the archmage of Keoland has failed, and been turned into a dretch (or whatever the lowest demon is). The Lanthorn has been taken by the denizens of the demonweb, which the party must recover. Once recovered, they can use the Lanthorn to face Lolth, and either kill her or (more likely) force her to abandon the Demonweb, as well as her plans for Oerth.
Adventure Path:
Village of Hommlet: levels 1-3
Temple of Elemental Evil: 4-8
Lost Caverns of Tsojcanth (with Forgotten Temple of Tharizdun): 9-10
Against the Giants: 11-13
Descent into the Depths of the Earth: 14-15
Vault of the Drow: 16
Queen of the Demonweb Pits: 17