Where's your friend from? That might change the priorities.
As I used to live in Harlow, I have a northeastern bias...
Rainham, just a short walk from Purfleet Station, must be the best place to go in London. Some nice wet fields, good chances of Peregrine, Water Pipit along the Thames and more than enough gulls. Very well visited, so uncommon stuff gets found on a regular basis.
The Lea Valley has Goosander (Flood Relief Channel at Lower Nazeing, Holyfield Lake or (esp.) the smaller lakes between that and Broxbourne), Smew (best chances probably Amwell, Holyfield, Hooks Marsh) and Bittern (harder now because there is too much habitat, but Amwell is worth a try). I personally would not really bother with Rye Meads... I am less familiar with the locations in London proper (Girling Reservoir requires a good scope for the Black-necked Grebes wintering there). The reservoirs near Blackhorse Road Station might be worth a try. These places are all free, unlike the London Wetland Centre (which I never visited), but reaching some of them could be a bit troublesome.
Broxbourne Woods can have Crossbill, Hawfinch and Lesser Spotted Woodpecker esp. around Danemead, but it is rather hit and (more) miss! I never really bothered with Epping Forest, except Connaught Water which is good for escaped waterfowl...
Further afield Elmley etc. are indeed good options, as could be Southend Pier.
In London itself, I'd give the parks a miss and head for the museums instead!