I am not sure if anyone has done it but it should be possible with a PICAXE. The main issue would seem to be constructing the HUD itself, getting a display to appear as if floating in space by whatever means is used.
Once that's solved any display which the PICAXE can drive should be usable, text or graphics, though there may be issues with update speeds for larger graphical displays.
Small OLED displays may be good for a HUD because they are quite bright with well defined pixels.
128x64 OLED displays like the SSD1306 can have their complete display updated by a PICAXE at a 50 fps rate, so reasonable update rates should be achievable if reading frames from I2C Eeprom or when updating an in-memory copy of the display and churning that out.
Another option may be an SBC with an attached display, tablet or phone, which can format and display data sent to it from a PICAXE, off-loading the heavy-lifting to that.
How practical any solution would be probably depends on type of HUD required. Desktop or in-vehicle use would probably be easier than something head mounted. There will probably also be regulatory and safety issues to consider in some cases.