my 2 cents:

I would bring a small bottle of diluted shampoo, which I would empty into a plastic cup of water, then dump this on my head in the sink and rinse with more cups of water. It was just enough shampoo to do something and enough water to rinse. I would also bring a small hand towel for drying my hair and body.

For my body I would use diluted soap and an actual wash cloth in the key areas such as face; chest; armpits and crotch. I wold shower at home first to reduce the need for a thorough washing.

I would change all my clothes from head to toe; including socks underwear, etc. I brought a small plastic bag for the laundry and zipped that up in my rack trunk away from people.

I would stockpile clothes and food at work but always brought my laundry home every day and used fresh underwear and sicks everyday. Pants or shorts and some shirts can be reused. I like to wear shirt liners or t-shirts and change them every day.

It was a lot of work with or without a shower. I had access to a private bathroom and I also arrived at least an hour before anyone else.

For a couple of years I exercised at lunch too and had to redo the whole sequence mid-day. Yeah that's a lot of laundry to carry on a bike but I only commuted in the summer and wore as little as possible - very casual office.