Let's take a look at some tips for the London accent, or what we call estuary. Estuary is a river tributary off of the Thames, so think of this accent as sort of off chute of the RP accent. Take a look at this same oral posture for RO. The jewel is raised and forward just a little bit. The tongue is raised in the mouth, but not pulled back. You get more glottal stops like in the word department. In RP we would say department, but in estuary we would say department. You get more vocalized "L" sounds.
In RP we'd say milk. In estuary we might say milk. In the word football, in RP we might say football. In estuary we might say football. Hear the difference there? Just a little bit more glottal, a little bit more urban sound. It's not all the way to Cockney. It's not like football or milk, but it's on the way there. If you think about all the British accents at a spectrum, you've got RP on one end and Cockney on the other. Estuary is somewhere in the middle.
Just like in RP, is estuary we would look for elisions. We'd look for words that end in an "R" and the following word begins with a vowel so you hear that "R." My brother owns a number of them. My brother owns a number of them. You wouldn't say my brother owns, you would say my brother owns a number of them.
In estuary, you might see some vestigial "R" action going on. R's get thrown in when they really shouldn't be there at all. In the phrase, "Rebecca and I saw each other yesterday," you might hear someone speaking that London estuary accent. "Rebeccer and I sawr each other the other day." Unlike the RP accent, there are no tact R's in estuary. You wouldn't say very in estuary. You would say very.
There can be some H dropping. Things that you may consider very Cockney sounding for your modern London accent, you want to drop the H's. "I went home for holidays." Sounds perfectly fine. In RP it would be, "I went home for the holiday." In estuary it would be I went home for the holiday.
What's the musicality of the estuary accent? It's very quick. It's very musical. You use pitch to make emphasis and link the vowel instead of loudness. Don't take my word for it. Go and listen to some modern London speakers and watch their mouths. Listen to the way they speak. Pick up some tips of your own.