I really hope you’re on the train OP.
For what it’s worth, I used to work in a West End box office and people came on their own all the time, it is a very common occurrence to sell single seats to business people/tourists etc. The staff won’t bat an eyelid and the other audience members won’t know, everybody is so wrapped up in their own experience that they just won’t notice or care.
Also, Wicked is fab and you’ll kick yourself for missing it, especially if you’ve got good seats.
As for the shopping trip. If you can go to the supermarket on your own, you can go shopping in London on your own, there is reallly very little difference apart from the fact that it’s unfamiliar territory. Transport is a piece of piss (and I promise that if you get a bit confused there are always people to hand who will help if you pop into a shop or cafe. Don’t ask people on the street, only because they are likely to be tourists too and won’t know and you may get disheartened).
Lots of people will be pottering about on their own, having lunch on their break blah, blah, blah - just take a book or your phone with earphones or mumsnet, honestly a person eating alone in London is utterly unremarkable, there is no need to feel self conscious at all.