When will Ubuntu 20.04 upgrade the kernel version?

The first HWE kernel will ship with Ubuntu 20.04.2, currently no date for that is available (see https://wiki.ubuntu.com/FocalFossa/ReleaseSchedule) but it'll occur after 20.10's release as it'll be 19.10's kernel that gets used.

Ubuntu 20.10 is scheduled to be released 22-October-2020 (https://discourse.ubuntu.com/t/groovy-gorilla-release-schedule/15531) so it'll be after that.

It is very unlikely that Ubuntu will ever have the 5.5 kernel version.

Ubuntu 20.04.2 will have a new HWE kernel, but it will be released in a few months, closer to the year end.

Update: Ubuntu 20.04.2 will have the 5.8 kernel.

According to the Ubuntu wiki:

The 20.04.2 and newer point releases will ship with an updated kernel and X stack by default for the desktop. Server installations will default to the GA kernel and provide the enablement kernel as optional.

The point release 20.04.2 is currently scheduled for release on February 11th 2021.

For people who do not wish to wait until then, it is possible to install the HWE stack manually before this date. On the date of writing this (Jan 7th 2021), Kernel 5.8 is available in the HWE stack. As mentioned in the first link above, it can be installed in the following ways:

On desktop systems:

sudo apt-get install --install-recommends linux-generic-hwe-20.04 xserver-xorg-hwe-20.04  

On server systems:

sudo apt-get install --install-recommends linux-generic-hwe-20.04