Of course you can! If anything, Friday is the perfect time to do it.
It's actually pretty common in the UK to see what appears to be men cross-dressing in public; it happens every single weekend in every major city, thanks to stag dos! It's also very common with charity events like Red Nose Day. People are going to react like it's all in good fun. They won't realise that this is something serious to you, so it's sort of like going stealth: you could test your femme mode in public without the public realising that that's what you're doing. It's a good way to gauge people's reactions to you en femme, and you might learn some useful things.
So in the context of the show, you could say you're doing drag in a drag Beetle and use it as an angle with which to raise money. This is entirely appropriate in British culture, and you'd get a positive response. And if anyone challenges you (e.g. in the services) you can just say you're doing drag in a drag car for Red Nose Day; would they like to donate? and any awkwardness would be defused.
Seriously, in the UK this would be a brilliant way to take Davina out for a spin (literally!) in public and it's probably one of the few times in the year when anything goes, so you'd be able to do it with minimal hassle. Do be aware though that British people tend to think it's hilarious to see someone they think is a man in a dress apparently having a laugh; they think they're in on a joke. So as long as you're able to not take that personally, you'll be fine.
And well done you for doing this: it's wonderful of you to raise money for the homeless.
