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Ex-LDN sheffielder here!
Sheffield is great - pubs are great, peaks access is great. Its a bit of weird place in that the city centre is not exactly thriving (though new development is happening) but lots of the suburbs are really really good with lots of stuff happening.
We initially moved to Nether Edge and I'd highly recommend there - a peaceful enclave between the more grimy pubs/bars/bakeries/kebabs of Abbeydale and London Roads, and the mix of the students and posh of Sharrow and Ecclesall Road.
On the best side for the peaks as well, escaping up Ringinglow Road. Or off road, can ride from Endcliff Park, out along the Porter Brook up the Mountain Bike tracks at Lady Cannings completely off road.
Nice! I used to ride through endcliffe pk all the way out to the peaks to MTB when I lived on brocco bank. Round Sharrowvale is a good shout, nice village feel round there. Am envious!