Some of the advice is a little surprising.
These names are normally ˈɔːldwɪtʃ, ˈbʌrə, kəˈdʌɡən, ˈtʃɪzɪk, ˈklæpəm, ˈdeʔfəd, ˈdʌlɪdʒ. RP usually distinguishes ˈɔːl from əʊl and ɒl (so that Paul ≠ pole ≠ Poll(y)), and Aldwych has ɔːl or possibly ɒl, but not əʊl. Of course many speakers have the GOAT allophone ɒʊ when dark l follows, as here; and Londoners tend to vocalize dark l, making cold kɒod; but I had thought that most would not merge the result with the ɔːo of l-vocalized called. Hence I am surprised to see Aldwich explained as “old witch”; though I suppose “all’d witch” or “auld witch” would be orthographically awkward. (Anyhow, the main point is that -wych stands for wɪtʃ, not wɪk.)
With Clapham, the basic ˈklæpəm can of course be reduced to ˈklæʔm̩ by the regular processes of syllabic consonant formation and glottalling.
The final consonant in Dulwich is, in my judgment, more often dʒ than tʃ, though both are possible; it’s odd that the anonymous author should prescribe the voiceless affricate in Dulwich but the voiced one in Greenwich and Woolwich, where the same hesitation between the two possibilities for -ch applies.
That’s ˈhəʊbən, ˈhɒmətən, ˈaɪzəlwɜːθ, with the usual syllabic-consonant options, plus possible glottalling in ˈhɒməʔn̩ and weakening in ˈaɪzl̩wəθ (or, of course, a more London-y ˈɑɪzowəf). Initial h is just as likely to be dropped/retained in Holborn as in Homerton.
So, ˈplɑːstəʊ (though I’ll allow people from the north of England and the Americans to say ˈplæstəʊ if they prefer), ˈrɒðəhaɪð, ˈraɪslɪp,ˈsʌðək, ˈstretəm, ˌθeɪdənˈbɔɪz, ˈtɒtənəm, ˈwɒpɪŋ. Regular optional processes generate the variants ˈrɒvəhaɪv, ˈstreʔm̩ and ˈtɒʔnəm; there is also an archaic variant ˈredrɪf (Redriff) for Rotherhithe; and if you drop the h in the usual form you'll get an internal linking r, ˈrɒvəraɪv. The Cockney tube train driver on my AofE recording pronounces his part of London, Wapping, as ˈwɒpʔɪn.