Plexie, which 3 books did you get from Persephone?
Are you coming in by train. from Euston? The London Review Bookshop is near Persephone and is huge.
If you and your birthday money get as far as Charing Cross Road, Any Amount of Books near Leicester Square tube station is a sort of secondhand shop, though actually a large amount of the stock is review copies of quite recent publications - around £3-£4 for paperbacks and more for hardbacks, though a bit pricy often for new non fiction.
They also have a section of £1 books outside and in the doorway - some funny little bargains from the 80s (like Virago's more contemporary publications of the time) and some books which are more commercial than literary as they tend to focus on literary fiction (with a crime section and a small SF section).
The Oxfam Bookshop and Bookmarks are across the road from each other on Bloomsbury Street, and if you look at the map you could go there on the way to Persephone and the LRB or on the way back, particularly if you run out of time to travel too much further south or west. If you go to Persephone first, you can do LRB and Oxfam quite easily with less than 2 minutes turning off towards each, then progress towards Tottenham Court Road tube station and turn down Charing Cross Road for Foyles/Any Amount of Books. For Waterstones you turn west along Shaftesbury Avenue.
This all involves a bit of traipsing around but it's really not worth walking to the nearest tube and getting down to the platform for a journey that is only one or two stops in central London, especially on the Piccadilly Line. In some cases you will actually spend more time and energy just getting in and out of the stations than you would staying above ground - eg Persephone to Russell Square, Piccadilly Line to Leicester Square to Foyles would involve 10 minutes walk above ground, getting in and out (and lots of escalators and steps at LS), and it's about 25 minutes walk above ground from shop to shop, perhaps less, with the chance to do others on the way, on foot. If your feet are tired you can do some of this on a bus but they move slowly through traffic. Similarly Waterstones' flagship branch is nearest to Piccadilly but it's actually between Leicester Square and Piccadilly and is probably less than 10 minutes walk from Foyles if you don't get distracted on the way. But if you're find that you're short of time and don't want to miss Waterstones out, the Gower Street branch is handier for being in the Euston/Bloomsbury area.
It's only worth getting on the tube if you want to venture up to the Owl Bookshop in Kentish Town (Northern line from Euston/Tottenham Court Road/Leicester Square, depending again on the order you plan to "do" shops in, a bit out of the way compared to the others. It's now part of Daunt Books operations but still has its own flavour as a large formerly independent bookshop which often has books it's difficult to find anywhere else as well as the books you can find back in Manchester, eg when I had a book token and wanted to buy some specific Virago Modern Classics reprints by Rumer Godden, Owl had more of them than even the flagship Waterstones store.