I'll defer to MετάEd, though I hope he realises that many consider the Yorkshire - Lancashire divide unsurmountable (isn't that a mixed metaphor?) - especially the Woodhead Pass in the snow.
In Lancashire, expressions like 'He worked at t'mill' and 'She went down t'Mill't'mill' were once common. They're usually spelt out this way.
The pronunciation seems different from across the Pennines, though - more like
'He work təh mill' where the schwa is really almost a schwa and a half with a slightly raised intonation on the trailing half
and
'She went dowən Mill'mill' where the schwa is really only about half a schwa
- no sign of the 't' from the 'the'., except in the variant where the preposition to isn't elided (but the 'usual' spelling is the same):
'She went down tə Mill' (again spelt She went down t'mill).