I've actually run the Delve, and had characters need to be replaced on several occasions. tl;dr: I'd have the new PC come in at level 14, with about the same XP total and wealth as the average party member.
For some context: the Delve is a 20-level megadungeon, designed to take PCs from level 1 to 20. Each level of the dungeon corresponds (more-or-less) to a character level. The levels are all themed (there's an "aberration" level, a "demon" level, a prison, etc.), with some small-but-useful connections between levels (e.g. a couple of fast-travel options to get from the top to the middle, keys on one level that open doors on the next). Levels of the dungeon are connected by stairs: when you find the stairs down, you can go down to the next level if you think you're ready for it. There is light continuity, mostly through repeated references to some NPCs. There are few, if any, ticking timers beyond the encounter level. Many encounters have a "revisit" section, with information on what is likely to happen if the PCs come back to a room days or weeks later.
I would strongly encourage that the replacement character come in at level 14 (and at about the same XP total as the rest of the party, so they can level at the same time as everybody else, and nobody has to deal with "different PC level XP reward" math). The challenges - especially on the lower levels - are designed with an APL equal to the level's ... level. I would also recommend the new character have starting wealth equal to the (rough) average of the rest of the party. The former will help ensure that the party isn't wholly missing access to resources that higher level encounters might assume are present and is the least punitive to the player; the latter helps prevent the new character from being over- or under-geared and either stealing focus or not being able to fully contribute.
Which, that's honestly what I do in about any campaign, but doubly so in pre-generated adventures.
Starting at level 1 will absolutely not work mechanically (either the party cake-walks through a lot of revisit encounters in the upper levels or the level 1 PC will either die upon rolling initiative, putting you right back where you are now, or the rest of the party will need to protect them, making them a drain on the party's resources rather than a contributing member) or socially (cake-walking back through the first 14 levels of the Delve would be mostly boring and/or tedious).
One option for getting the new PC up to level (and gear) could be to have the party cake-walk them through the revisit encounters off-screen. Doing so would also explain how the new PC knows stuff that the old PC knew (because the player knows it), and how they can use things like the teleportation altar down to level 7(?); maybe speed-run through it, letting the players reminisce about cool stuff they did for a session.
That said, I would actually be inclined to have a new PC start at 1st level (with high-level gear) if they were on level 19: a fair amount of that level is letting high-level PCs be awesome (e.g. fighting an infinite plane of kobolds), so a 1st level PC wouldn't be nearly the drain they would be in any other situation, and resolving the level solves the level disparity issue nicely.