a pronounced "t" sound for the more silent version (in words such as Britain, little, etc.) …I think what he is referring to is actually a change in the treatment of the nasal or lateral that follows the t, namely the replacement of the syllabic n or l by a sequence ən or əl respectively.
Hence instead of a nasally released plosive in Britain ˈbrɪtn̩ (the ‘more silent’ version) we get an orally released plosive (the ‘pronounced’ version) followed by a schwa, thus ˈbrɪtən. Correspondingly, instead of a laterally released plosive in little ˈlɪtl̩ we get a centrally released plosive likewise followed by a schwa, thus ˈlɪtəl (or further development to ˈlɪto and the like).
The same thing happens with d in this environment, as in garden and middle, where alongside traditional ˈɡɑːdn̩ and ˈmɪdl̩ you can also hear ˈɡɑːdən and ˈmɪdəl, ˈmɪdo. It’s not primarily a change in the plosive as such but a change from a syllabic consonant to a schwa plus nonsyllabic consonant.
This tendency to avoid nasal/lateral release has indeed often been commented on. People have sometimes characterized the result as sounding ‘childish’.
There is also a third possibility with the t in words such as Britain, little. That is the use of a glottal stop ʔ rather than an alveolar t before the syllabic consonant, giving ˈbrɪʔn̩, ˈlɪʔl̩. The former often seems to pass unnoticed; the latter may attract attention.
Cruttenden says (Gimson’s Pronunciation of English, seventh edition, p. 180):
Use of [ʔ] to replace /t/ [sic] … before syllabic [n̩] … was until recently stigmatized as non-RP but … [is] now acceptable in London Regional RP.He refers to Anne Fabricius’s article ‘Ongoing change in modern RP: evidence for the disappearing stigma of t-glottalling’, English Worldwide 23: 115-136.