By default yes, all of the worlds in 5e exist in the same multiverse

The spell Dream of the Blue Veil from Tasha's Cauldron of Everything explicitly sets all worlds as being in the same material plane:

You and up to eight willing creatures within range fall unconscious for the spell’s duration and experience visions of another world on the Material Plane, such as Oerth, Toril, Krynn, or Eberron. [...]

Beyond this spell, we have other evidence for this being the case. D&D 5e is set in a multiverse by default, which contains many worlds. This is explicitly called out in the Gods of the Multiverse appendix of the Players Handbook (PHB):

Religion is an important part of life in the worlds of the D&D multiverse.

[...]

Many people in the worlds of D&D worship different gods at different times and circumstances.

Other worlds are explicitly called out in the D&D Pantheons section of this appendix:

Each world in the D&D multiverse has its own pantheons of deities, ranging in size from the teeming pantheons of the Forgotten Realms and Greyhawk to the more focused religions of Eberron and Dragonlance. [...]

The Dungeon Masters Guide (DMG) also explicitly calls out that there are multiple worlds in the Material Plane in the Making a Multiverse chapter:

[...] In this context, the Material Plane is the nexus where all these philosophical and elemental forces collide in the jumbled existence of mortal life and matter. The worlds of D&D exist within the Material Plane, making it the starting point for most campaigns and adventures. The rest of the multiverse is defined in relation to the Material Plane. [...]

When talking about the Material Plane, this chapter also has a section called Known Worlds of the Material Plane which states (along with more in depth per world descriptions):

Worlds of the Material Plane are infinitely diverse. The most widely known worlds are the ones that have been published as official campaign settings for the D&D game over the years. If your campaign takes place on one of these worlds, that world belongs to you in your campaign. Your version of the world can diverge wildly from what’s in print.

Chapter 11 of the DMG A World of your Own it reinforces this idea:

This book, the Player’s Handbook, and the Monster Manual present the default assumptions for how the worlds of D&D work. Among the established settings of D&D, the Forgotten Realms, Greyhawk, Dragonlance, and Mystara don’t stray very far from those assumptions. Settings such as Dark Sun, Eberron, Ravenloft, Spelljammer, and Planescape venture further away from that baseline. As you create your own world, it’s up to you to decide where on the spectrum you want your world to fall.

If you needed more evidence of this, the adventure Dungeon of the Mad Mage (set on Faerun) makes named worlds of other campaign settings explicitly set within the same multiverse:

Yes, the multiverse is shared. However the inhabitants perceive the multiverse in different ways.

The answer by SevenSidedDie is reasonable. If you are looking for a semi-official answer, here are a series of Q&A tweets from Jeremy Crawford (January 22, 2015, emphases mine):

Does the #dnd tabletop RPG have one official setting? The answer is yes. That setting is the multiverse, which includes all #dnd worlds.

Q: my kid keeps asking me to explain whether each setting is a planet or a plane and I don’t know the answer.
A: The worlds occupy pockets of the Material Plane—sort of like planets but in a space shaped by magic and divine forces.

Q: Does this mean that there is one consistent cosmology/planar arrangement for all worlds? E.g 3e Eberron planes “wrong”?
A: The inhabitants of each world in the Material Plane have different ways of imagining the multiverse (check out chapter 2 in the DMG).

Yes, and also No, but sorta mostly Yes

The default setting of D&D 5e is “The Multiverse”, and as far as I can tell this contains all possible worlds, including words with different cosmologies. This is a result of The Multiverse being the default setting, and the canonical notes in the DMG that the planes can be chosen à la carte by the DM when designing a campaign setting.

The latter means that each setting can have a different planar cosmology, as designed by the DM. The former means that even such worlds share the same Multiverse, despite differing planar cosmologies.

The conclusion then is that no, the existing published worlds don’t have a shared cosmology… and also that there is another version of each that does.

Aside, another consequence of the Multiverse is that one group’s Forgotten Realms (to grab one example world) is a distinct part of the Multiverse from each other group’s Forgotten Realms, all of them part of the Multiverse. Basically, there are infinite variations of each world, with infinite variations of cosmologies, and infinite permutations of sets of these with shared and cosmologies and without.

The upshot: practically, players consider settings and cosmology to be somewhat shared

In practice, many groups appear to see the Forgotten Realms as one place by default, and seem to assume that there is one shared cosmology. Most people don’t seem to even need it to be clearly defined in the first place.

Consequently, you can write and publish with the confidence that most of your potential audience will be happy to accept the assumption that (e.g.) the Forgotten Realms and Greyhawk share the same (e.g.) Bytopia, if your product makes that assumption.

Put another way: the rest of them — that is, anyone who has opted to take the DMG’s advice and craft a separate cosmology —may still find use in cherry-picking your material to suite their needs just as they did with the DMG chapter on cosmology design.

You can even encourage this, by giving your default assumptions but taking the DMG’s lead and noting that everything presented is optional and can be mixed-and-matched. Even if your assumptions make that hard to do, you should be fine — anyone doing an à la carte treatment is already confident adjusting your material for you.