A good samaritan created a PPA which you can add and install GNOME 40 in Ubuntu 21.04 (Hirsute Hippo). And this is only for testing the packages.
Warning: Use this PPA and add it in your system at your own risk. Things shall break. So do it for experiment only.
GNOME 40 desktop recently released with many groundbreaking changes such as revamped Activities Overview, reworked workspaces, and many more. But unfortunately, the changes are too much for the Ubuntu team to consider for Ubuntu 21.04 Hirsute Hippo. Hence, officially it will not feature GNOME 40.
GNOME 40 in Ubuntu 21.04 [Testing Only]
But there is a way where you can install the latest GNOME 40 in Ubuntu 21.04 for testing purposes. A PPA is created by a user and as per this Reddit post, it seems is working well in Ubuntu 21.04.
This is how you can do it.
Make sure you have Ubuntu 21.04 installed. If it’s not, grab the .iso here.
Open a terminal and add the unofficial PPA.
sudo add-apt-repository ppa:shemgp/gnome-40
sudo apt update
Then install the GNOME window manager mutter from the repo.
sudo apt install mutter
You need to separately install gnome-shell
from the PPA to get the latest GNOME 40 changes.
sudo apt install gnome-shell
And you need the latest gnome-session
as well because the Ubuntu Yaru theme is not yet properly ported to GNOME 40. So the Ubuntu session won’t work.
sudo apt install gnome-session
Do a system upgrade to make sure get all the GNOME 40 applications via PPA. Remember, this will only upgrade the packages.
sudo apt upgrade
After the above steps are complete, you may want to check the GNOME version via this command. It should show GNOME 40.
gnome-shell --version
Finally, reboot your system.
reboot
In the login screen, choose GNOME Session, not the Ubuntu Session. I would recommend use GNOME on Xorg. Don’t even try the Wayland for the unknowns!
If all goes well, then you should get the latest GNOME 40 in Ubuntu 21.04.
Couple of things I noticed when I first boot it up. The Ubuntu color palette and GNOME’s grey doesn’t look nice. The system-wide fonts are different as fonts-cantarell is not installed. However, functionality-wise GNOME 40 working great to my surprise!
Here are couple of screenshots.


Closing Notes
If you really want to get a proper GNOME 40 experience, try with Fedora 34, or install the packages via Arch Linux. To get it officially via Ubuntu, you may need to wait until the Ubuntu 21.10 October 2021 release. So until then, you can experience it right now via the above method. Another word of caution, do not try this PPA in Ubuntu 20.04 LTS.