I emailed Toyota today, partly to ask if I could instruct the car not to be washed, but also to see if I could get a status update, as the status has been "Processing" for just under 2 months and I hadn't heard any more since. I got a reply telling me that "Our system still indicates an estimated arrival of your vehicle of September 2022 but please note this is an estimate, it can go forward as well as back."
Hopefully that means things are on track and I can expect to see it go into build fairly soon.