What do you expect "afternoon tea" to be? It is just a snack of tea and cakes, a very light meal, or small sandwiches, which is consumed around about 4 to 5 o'clock to fill in the gap between late breakfast and dinner. Some places (usually the expensive hotels or restaurants) have tried to blow it up into some sort of exclusive aristocratic ritual which excludes the "ordinary mortal".
If you stick to that assumption then you will inevitably be disapointed when you tuck into a small fairy cake and cup of tea then realise that you have paid over £20 per person for the privilege of having it in a fancy tea room.
There are hundreds of places in London where you can have good quality afternoon tea without booking. If you want to stick to the big hotels or posher restaurants then you will find that they fill up, so there may be a problem if you just "turn up" that's why they like you to book.
Why not forget stereotypes of London (I realise this can be difficult) and just wander into somewhere that is not "international"? Forget the Ritz or Savoy (which has made a feature of publicising its "afternoon tea") and other large hotel chains. This means look at smaller hotels or even restaurants in places where you wouldn't think of looking.
Places like Fortnum and Mason (a very prestigeous "royal" grocer); Harrods (the biggest department store); the Victoria and Albert Museum (has a beuatiful tiled tea roon) all have excellent restaurants where you can meet and enjoy tea and cakes. You can find plenty of smaller restaurants and cafes in the Covent Garden area where you can enjoy great quality afternoon tea in historic surroundings without being "ripped off". Booking is not done in these places so just turn up - if one is full then try nearby.
Look at these links for some guidance and hints:
http://www.timeout.com/london/restaurants/the-15-quirkiest-afternoon-teas-in-london
http://www.telegraph.co.uk/travel/destinations/europe/uk/london/9124793/Londons-ten-best-afternoon-teas.html