From reading the manual I know this works with the om2 on automatic, but will it work (other than on manual) with an om30, 40, or om1?

Your question is slightly confusing, but I'll do my best.
The T32 (and T20 and T45) flashes incorporate their own auto flash sensor and circuitry. So for any camera - OM or not - you can flip the back panel over to reveal the ASA, distance and f/stop information and slide the switch to one of the auto exposure ranges, or set it to manual and use the guide number scale or a flash meter to set your camera's aperture.
In addition, for those Olympus cameras that offer off-the-film flash metering, you can flip that back panel over to the almost blank side, and rely on the in camera off-the-film flash meter.
All of the OM bodies I use offer off-the-film flash metering - OM2n, OM 20/G, OM2s and OM4T. Same for the OM2 and OM4.
The OM1 does not offer off-the-film flash metering. Nor does the OM1n.
I don't have enough experience with the OM30 or OM40 to answer your question about them.

Thanks, I don't think it works with TTL on the om40. Not too bothered as I prefer natural light. Just surprised that a 'newer'' model of an OM wouldn't make use of TTL.

The OM 30 does not but the OM40 will operate TTL flash on auto

Thanks I re-read the OM40 manual properly this time and I can use the T32 on auto. The manual that came with the flash was probably!y printed before the 40 was made and so didn't mention it.