You can create a Doppler effect using the “Sliding Time Scale/Pitch Shift” effect (included in Audacity 1.3.13).
It’s a little bit cumbersome to do as you will need to process the audio in sections.
The “coming toward you” section needs the pitch to be shifted up a little and the speed (Tempo) increased by the same %.
The “going away from you” needs the pitch to be lowered a little and the tempo slowed down.
Make a note of the settings that you use for these two sections as you will need them for the next section:
The “going past you” section needs to “slide” from the quicker/higher settings to the slower/lower settings.
As well as the Doppler effect you need the volume to gradually increase as the siren approaches, and reduce as it goes away - use the “Envelope Tool” or “Fade” effects for this.
If you use the Envelope tool you will need to “Render” the track before adding Reverb (Tracks menu > Mix and Render).
To produce a realistically changing reverb, make a duplicate of the track (Ctrl+D) and apply the reverb to the duplicate track. You will then need to “shape” the volume of each track so that the distant siren has relatively more reverb than the close siren. When the siren is very close the duplicate (reverb) track should probably be silent (or relatively low) so that you have only (or mostly) the direct (dry) siren sound.
In addition to the previously mentioned reverbs, there’s a new one here: https://forum.audacityteam.org/t/stereo-reverb/20310/1