OP, do you mean nice as in:
A - tree lined streets, houses in excellent condition with beautiful gardens, shiny new cars frequent, old bangers virtually non-existent, many residents may consider private schools for their kids.
or
B - houses are in good condition (with the occasional scruffy one) but not necessarily as large and lovely as the above, often have ok sized gardens, some shiny new cars but old ones equally common, nice community feel, feels safe for kids and not rowdy at night, most residents would not consider private schools as they couldn't afford it.
My perception of nice in Bristol is that Clifton, Westbury on Trym, Redland etc fall into category A. Totterdown, Redfield, Horfield, St Werburghs and the like fall into category B.
My minimum standard for "nice area" is "I don't have to worry about meeting a drug dealer or prostitute on every corner, there aren't regular fights outside local pubs, I don't feel intimidated walking home at night and I don't think anything of popping out to the corner shop at 9pm for a pint of milk". Both the above meet that, but there's a fairly big difference in your monthly rental!
FWIW, we managed on similar take home to what you're anticipating during mat leave (obvs no nursery fees then) while living in a B area, and we could probably manage with one of us at home (and thus able to cut out nursery fees and shop a bit more cheaply/cook more at home etc). But it would be tight. We don't have a 3 bed, but checked Rightmove and there are 3 beds for the same cost as our monthly mortgage (about 900 pcm) in Horfield, for example. Our monthly take home is about £4k and we are pretty comfortable on that (with nursery fees + mortgage and small commuting costs). Can eat out, save, don't have to worry about every penny.