Easy peasy and this will clean up your teaspoons also.
Any clothes washing powder, large pile into mucky cup, add boiling water, stick in the tea stained teaspoons, leave it fr a bit - hour will do, longer if you like. Then rinse off and wipe the grime away. Worse it is, longer you should leave it, and if you use the green part of a scourer, they will be shiny shiny.
Have fun.