Tube tops are great because you can turn them into a scrunchie or cowl scarf in a pinch.

I was going to suggest something disposable, like a garbage bag, newspaper, or an old teatowel or length of fabric held together with a peg or tied in a knot. Might not look great but you can throw it away once you arrive. But then I guess the problem is you can't use it for the trip home again? I wonder whether you could stash normal clothing somewhere out of the way, e.g. at a train station or hidden in a garden somewhere. Maybe it'll still be there later when you return? Not sure what time you would be heading home again or if you would use public transport for that, but I'd be surprised if those sorts of indecency laws were enforced late at night, it seems more like a daylight hours thing.

I would suggest getting 'Nutter to drive us in, but it's going to be hectic out there, and ze doesn't really like too much traffic.

I'm sorry I can't really answer your question itself; I'm really not qualified to, but could you wear a long sleeve shirt/light jacket, then tie it around your waist when you're done with it? Or would that still be in the way?