Sit down and drink cold water until you quit sweating. Find restroom. Cold water sink shampoo, fully dressed in bike clothes. Dampen two washrags in cold water. Apply a dab of soap to one.
Move to handicap stall and strip. Use soapy washrag to completely scrub everying starting at the neck and working down. Use clean washrag to re-wipe all areas to remove any soap residue. Dry off with big towel, apply toiletries, dress in clean clothes. Clean up all splashes so nobody can tell the difference, be considerate of others.
I had a clean-up kit that I kept at work - a 2-quart bucket, 2 microfiber washcloths, 1 XL camp towel (microfiber, thin and dries fast), a sham-wow to get the hair almost dry. The bucket was just for clean rinse water for the wipedown rag, the soapy rag never went into it. The soap was liquid antibacterial (Dial) with a fresh herbal scent. The toiletries were all in a ziplock bag, sample-size containers that I refilled at home from big bottles every now and then. The towels got a few uses before they went home to be laundered.
Honestly - after a cool water cleanup in an air-conditioned well-ventilated bathroom, I was as clean as if I had taken a shower and it was just as fast if not faster. A handicap stall has enough manuevering room and is often cleaner than a regular stall. Generally bathrooms are vacant in the morning before work.
I keep a pair of shoes, a cardigan sweater and a belt at work. I only buy clothes that don't need to be ironed.