You will mostly be missing jokes referencing events and characters.
The majority of the missing context will be jokes and inconsequential references to old characters and historical events in the world's timeline. This was the case in Campaign 2 (and has so far been the case in Campaign 3); occasionally one of the cast would crack a joke about how a Vox Machina character would have acted in a particular situation or something like that, or one of them would remember something and Matt would remind them that was something that happened to their Vox Machina character, not their Mighty Nein character.
Matt is very intentional about not relying on past characters too much.
In the first episode of 4-Sided Dive, the successor to talk show Talks Machina, Matt gives some commentary into how he has approached integrating characters from past campaigns into Campaign 3. This episode aired after C3E18, so it contains spoilers up to that point, as well as spoilers for ExU: Kymal. Here are some relevant quotes that give some insight into this question:
Robbie: I have a question. When you're tying these threads together as a player and DM, and you guys have spent seven years building this lore and creating canon through your publications and the games, and all that. I know that fans in general love tie ins and they love nostalgia. That's just one of those things I think is inherent in people. But, I feel like, and something you;re doing well, I wonder how you naviate that razor's edge of giving the cool tie in for the fans, but also walking that fine line of not getting into the realm of hokey. Because you could, leaning on it too hard, we love Vox Machina, look how it's tied! But, it's gotta be perfect, and I feel like those little seeds you guys are planting are cool, because it's a canon world that has a timeline, and you planted the seeds, and now you're gently harvesting them throughout this universe. What's that like? What does that discussion look like?
Matt: That is the challenge. I want to make sure we create something that doesn't harken too strongly on an existing story to where it robs your character of her own unique narrative. We don't want to have a group of unique characters and then the "Vox Machina tie in character". So it was very much like finding that balance where it is fun to do nods and sit in that world with realistic roots, because Vox Machina did leave a big impact on the world. But we definitely don't want any of those past elements to overshadow it. Let's ensure this is your story and your narrative.
So it seems Matt is being very intentional about keeping the narrative of Campaign 3 largely self-contained, even if some character backgrounds do involve characters from previous campaigns.
It may be worthwhile to catch up on Exandria Unlimited as you follow Campaign 3.
Exandria Unlimited is an immediate prequel to Campaign 3, and (minor spoiler):
Three of the characters of Campaign 3 begin their journey in Exandria Unlimited. So to get the details of who those characters are and why those characters appear in Campaign 3, you would have to get through Exandria Unlimited. However, without that context, you aren't missing too much, as it is briefly explained why they are there, and that would be far more context than you have for the other characters.
Further, we learn (minor spoiler for Exandria Unlimited):
A certain character from Campaign 1 is part of the backstory of a Campaign 3 character, and plays a role in the time between the end of Exandria Unlimited and the beginning of Campaign 3. If you aren't at least familiar with the characters of Campaign 1, you might miss some of the context of that Campaign 3 character's backstory.
Overall, I think you should be just fine to start with Campaign 3, especially if you catch up on Exandria Unlimited as time permits (8 episodes, roughly 32 hours of content). If you don't intend to actually sit through all of Campaign 1 and 2, you can just follow up on the lore threads from those campaigns as you encounter references you are unfamiliar with in Campaign 3.
The Critical Role website has a helpful article, "What you need to know before watching Exandria Unlimited", that suggest that Exandria Unlimited is a good place to start for those new to the series:
First things first! You don’t really need to know anything before you watch Exandria Unlimited. You can jump in completely fresh! Some of our players will be journeying into Exandria for the very first time as well.
So you definitely don't need anything from Campaigns 1 and 2 prior to Campaign 3, but Exandria Unlimited may be helpful for providing some of the context for the beginning of Campaign 3.
For a broad overview of the history of Exandria, watch this video: Exandria: An Intimate History (Campaign 1 spoilers). Replays of Exandria Unlimited can be found in several places:
Update through Episode 6, spoiler block contains spoilers
We learn in episode 6 that a character's backstory relates heavily to a certain antagonist from Campaign 1. Watching Legend of Vox Machina on Amazon Prime should get you up to speed without having to watch all the live play episodes from that part of Campaign 1.
Laudna’s Warlock patron is Delilah Briarwood, who she occasionally has conversations with in her mind.