Hi,
"writer" doesn't sound like "wider", it sounds like "rider".
Anyway, T's are sometimes tapped. It usually happens when T's are between two vowels and the following vowel is not stressed at all (not even a secondary stress on it). Example: Be
tty bough
t a bi
t of...
It often happens in other cases too, for example when the following sound is not really a pure vowel sound, but you could say it's "almost" a vowel. Examples are L's and R's. Examples: be
tter, hea
ter, bo
ttle, li
ttle.
W-sounds and Y-sounds don't lead to tapped T's.
It was ---> I(t) was - the T is not released/glottal stop
about you ---> abow chyoo - T and Y form a kind of CH sound.