There are lots of great birding sites in London, such as WWT London Wetland Centre
and RSPB Rainham Marshes
. Rainham is a coastal wetland with many reedbed trails, lagoons and hides, plus areas of scrubland. In June you can easily find warblers (reed, sedge, cetti's, blackcap, willow, chiffchaff) and a number of wildfowl and wader species. There is a common kingfisher hide where you can see them at the nest, and reedbed feeding stations for bearded tit and reed bunting, plus feeders outside the visitors centre which attract the common woodland species. Walking along the sea you can scan for gulls, rock and water pipits, common sandpiper, and birds of prey such as kestrels and marsh harriers. WWT London Wetland Centre is smaller and has less variety but is a good place to get started. One highlight is the sand martin hide where you can watch sand and house martins swooping over the wader scrapes.
is a good urban park to get familiar with UK birds, if you're new to the country, with excellent close-up encounters with grey herons, grebes, rails and ducks, and woodland birds.
) is nearby and has roosting Little Owl and Tawny Owl. Harder to find in the summer, but I can give you details about what trees they can be seen in.
I don't think Nightingale occur anywhere in London - the only place I've seen them is at RSPB Minsmere on the Suffolk coast, but that requires a whole day trip (a bit under 3hrs from London).
If you'd like any details or a guide/birding pal to show you around, do let me know and I'd be happy to help.