How to install uv when I already need an virtual environment to install it?

With the standalone installer, or any other method - pip, cargo, docker or just by downloading the binary. All is described in the documentation, see https://docs.astral.sh/uv/getting-started/installation/ .

Isn't this a chicken-egg-problem?

No. uv is written in rust, it doesn't require any virtual env or python.

How do you do it?

I just do pip install --user uv