I've been cycling commute for almost 30yr now, some places shower and others no.
Most important is to have towel and spare work cloths, wash in the toilet and off you go...
I think as long as you are putting on a change of shirt, shorts and socks when you get there, a wipe around the dark corners is quite enough.
People get overly paranoid about personal hygiene it seems to me.
there is this stuff + baby wipes..
When I was riding to work our place got flooded and we lost use of it including all the showers - new temporary place didn't have any but did have sinks and I bought a push on shower hose that fitted over the taps. Surprisingly effective, you could lean right over the basin and get your top half well and truly rinsed without making too much mess on the floor. Car to work once a week or so and clean shirts etc came in then.