There may be aspects of her courtship Mrs Elton would rather not share with the good folk of Highbury.

Maybe the money for Mr Elton's carriage came out of her bride-price, and(as he could not access the funds until after the wedding, and could not obtain (more?) credit in the time frame), she was obliged to be driven to her wedding in a rented chaise. I could imagine Mrs Elton hesitating before revealing the sordid details of her future husband tapping her up for cash, or even simply the indignity of being wedded from a rented carriage, with Mr Weston.

Mrs Elton does seem to avoid discussing what her home and family in Bristol were like.
When Mrs Elton speaks of
" Mrs. Jeffereys -- Clara Partridge, that was -- and of the two Milmans, now Mrs. Bird and Mrs. James Cooper"
and how " Selina said, when she has been going to Bristol, 'I really cannot get this girl to move from the house
it does seem as if she wanted to avoid returning to Bristol at all hazards, and worried that all her friends were getting married before her.
What was wrong with Bristol, is not explained. I imagine her uncles house would be a rather good one, as it is in the heart of Bristol and her sister does not share her aversion to it. Even if her fortune was made from human blood and misery, her father was dead and her uncle was not in trade. Bristol itself seems to have been a lively enough place - but Miss Hawkins seems determined not to settle near her home, and Mrs Elton determined to confine her reasons for settling in Highbury to the charms of her Caro Sposo.