@J.A.Streich's answer is great. I'm going to come at the question from a different direction.
I'll assume that the clone's maturation is complete for the duration of this answer.
What happens after Clone activates?
One key thing to remember is that 5e doesn't have hidden rules; in this particular case, that means that the spell does what it says it does:
This spell grows an inert duplicate of a living creature as a safeguard against death. This clone forms inside a sealed vessel and grows to full size and maturity after 120 days; you can also choose to have the clone be a younger version of the same creature. It remains inert and endures indefinitely, as long as its vessel remains undisturbed.
At any time after the clone matures, if the original creature dies, its soul transfers to the clone, provided that the soul is free and willing to return.
The clone is physically identical to the original and has the same personality, memories, and abilities, but none of the original’s equipment. The original creature’s physical remains, if they still exist, become inert and can’t thereafter be restored to life, since the creature’s soul is elsewhere.
Therefore: when the caster dies, they immediately wake up in their cloned body.
Can the clone make its way to the party even though those actions can’t really be discussed?
While the clone is in waiting, it's a helpless, inert bit of flesh and bone in a sealed vessel. Neither the vessel nor the clone inside have any particular protections against being damaged or disturbed. Due to that, it's exceedingly likely that the clone will be grown in a place that the caster has good reason to believe will be safe for a rather long period (120 days for maturation at the very least) while the caster is out adventuring - if they needed to babysit the clone, its whole purpose as "a safeguard against death" would be largely diminished.
Wizards can copy their spellbooks. Since the location of the vessel is believed to be safe, it makes sense for a wizard to leave a copy of their spellbook (or, at least, one with the most essential spells) next to the vessel. With their spellbook, they can always bootstrap the process of rejoining their party and reclaiming whatever gear might have been lost.
Regardless of the casting class of the caster, it would also be reasonable to store a few essential items next to the vessel - a few essential scrolls, wands, staves, and/or other magical (and mundane!) gear. This is unlikely to be a full replacement set of equipment, and may be specially chosen to aid in rejoining their party or recovering their "main" equipment. It seems to be the case that dying doesn't affect what spells a caster still has prepared, so they could start the process very quickly.
Does the party have to gather up the caster’s now-soulless corpse’s belongings and venture back to where the clone is stored so that the caster can rejoin the party?
IME, adventuring parties almost universally trust their party members entirely (at least in important things: the rogue might steal anything that isn't bolted down and cheat at dice, but won't steal a copper from the party or fudge the loot distribution). Presuming that that's the case, the rest of the party probably knows where the clone caster's clone is kept (IME, it'd probably be in whatever they think of as their home base) and so would know where to go to rejoin their pseudo-fallen comrade.
The party likely has plans regarding how to handle the (usually) wizard popping into a clone. If not, they should probably talk about it before it comes up.
Of particular note: the clone "has the same personality, memories, and abilities", so the plans can include contingencies based on how the caster died or what the party's state appeared to be at the moment of death.
This doesn’t seem terribly convenient, despite the fact that the caster has just cheated death.
Clone is an 8th level spell that costs 1,000 gp to cast plus requires a 2,000 gp vessel. It takes 120 days for the clone to be ready to use, after which the caster's death causes the soul to immediately enter the (or, at least, a clone). The wording suggests that the vessel is re-usable.
Revivify is a 3rd level spell that's a bit cheaper to cast (300 gp vs. 1,000 gp + a 2,000 gp vessel), but requires a caster available to cast it within a minute of the target dying. It also (probably?) doesn't work if the body is missing the head or other essential bits. I think it's the easiest/cheapest/fastest way to restore someone from the dead, but it comes with some significant limits. The target comes back with a single HP, but there's no resurrection sickness, so that's nice.
Raise Dead is a 5th level spell that's also a bit cheaper to cast at 500 gp. It can be cast within 10 days, though the target returns with a single HP and takes penalties on basically all d20 rolls for a few days. Like revivify, it probably doesn't work if the body is missing the head or other essential bits.
Resurrection is a 7th level spell with the same 1,000 gp cost. It takes an hour to cast, and it can't resurrect those who have died from old age (which, by my reading, clone can). Targets also take penalties on d20 rolls for a few days after coming back. It can be cast up to 100 years after the death, though, so there's more breathing room to find a caster.
True Resurrection is a 9th level spell that costs 25,000 gp. It's basically resurrection but slightly better in some edge cases.
Reincarnate is 5th level spell with a 1,000 gp cost to cast. It takes an hour to cast, and it doesn't guarantee that you'll get the same body - or even the same race, though, so racial bonuses and abilities might get moved around in unfortunate ways.
Critically, all of those spells - aside from revivify take an hour to cast. Revivify assumes there's a caster "right there" who has a 300 gp diamond and a spell slot - and a turn - to use on the spell. Clone allows its caster to start acting more-or-less immediately upon death, albeit in a potentially-far-removed location. With the right scrolls and core emergency gear, the clone body could be back to the party in just a couple of rounds without anyone still in the fight needing to spend a turn (or a spell slot) getting them back up.
So, it's situationally-convenient. Clone allows the caster to get back in the fight relatively quickly if the clone's vessel's proximity has the right tools (eg., a scroll of teleport would be one of the first things I put next to my clone's vessel). However, there are situations in which the clone body getting back into the fight would be hard (eg., planar shenanigans) or when not having a clone would be preferable (eg., if the party has planned for casting revivify a few times).
The general schtick of wizards, typically, is to be prepared for any eventuality ("I have a spell for that!"). Clone is just one more tool in a wizard's toolbox to be prepared for unfortunate eventualities.