look at the octopus thread for suggestions of fishmongers.
I use Ashbys in Crimscot street for meat. They are predominately a wholesalers but will sell to indivuals. The office is open 3am-4pm. The meat is good and i order over the phone and pick it up, You do need to know what you want as they do not have a counter.
Butchers - Suffolk Meats, Great Suffolk Street. Superb famiy butchers, need to drive there from Tower Bridge area though.
I get my fish from my main man 'John The Fish' who has a stall on Tatchbrook Stret in Pimlico, just off Vauxhall Bridge Road. Not in SE1 I know, but 15 min drive and you can park for free on Saturday. Best fish I've ever had - got some lovely Black Cod this weekend, some squid and beautiful Mackrel.
It is great food at Borough Market, but it's at least 25% more expensive than other places. It's great quality I know, but some things are over-priced (they get the same fish from Billingsgate as other fishmongers). Credit Cruch food shopping it's not.
I tend to agree with JGarcia re price of borough market food. And whilst there may be good stall shops on fri and saturday, I need to be able to buy good quaility food (in domestic not catering quantities) all week and not just at weekends. Also I need to purchase my food at the begining of the day. I have had wonderful help from Booths and the small organic stall in the market when shopping around 9o'clock in the morning on fridays but try ask turnips or furness fish for anything that early and the tell you to come back when they are open. applebys fish are helpful even before they officially open but the fish often arrives too late for me.