Will GNOME 3.38 ever be in the official Ubuntu 20.04 repositories?
Solution 1:
No.
LTS releases are intended to be stable in the sense that the software does not change. The software on 20.04, with limited exceptions, won't be updated. That's the whole point of an LTS.
- LTS releases continue to receive security updates from the Ubuntu Security Team, and major bugfixes. These security updates include newer versions of web browsers. However, a whole new desktop stack is neither.
Let's take a look at an example: The Gnome environment from 18.04...
gnome-shell | 3.28.1-0ubuntu2 | bionic
gnome-shell | 3.28.3+git20190124-0ubuntu18.04.2 | bionic-security
gnome-shell | 3.28.4-0ubuntu18.04.3 | bionic-updates
gnome-shell | 3.36.1-5ubuntu1 | focal
...see how gnome-shell 3.28 was in Ubuntu 18.04? You can see the minor updates from security patches and bugfixes, but it's still on version 3.28. Meanwhile, 20.04 uses Gnome 3.36. And two years from now, it's likely to still be using 3.36.
In general, if you want newer software, you should run a newer release of Ubuntu.
Solution 2:
Ubuntu 20.04 is an LTS release. Therefore, you wouldn't expect a lot in Software changes as it is meant to be stable.
However you can be sure Ubuntu 20.10 to be released on 22nd October will ship with GNOME 3.38.
If you really want to try out GNOME 3.38, Download img file for GNOME 3.38 Beta (approx 2Gb) From that Link and use it as a Virtual machine.