The big cities (and Scotland, Wales and NI) didn't vote this time, so the local election results won't necessarily be reflected in a GE.
I'm not sure that Brexit was an issue. I didn't hear it being discussed, although the reasons which underpinned voting in the referendum remain.
It's quite interesting that the Conservatives have lost hundreds of seats, but they haven't all gone to Labour by any means. Both the LDs and Greens have picked up a fair number of seats. That's not surprising in local elections. Firstly, people are concerned about local environmental issues, so vote Green. LDs have always been strong on local issues. Whether that translates to national voting is a different matter.
Secondly, people are disenchanted with the Conservatives, but can't bring themselves to vote Labour. In a GE, there will have to be some tactical voting for that to be reflected in the number of MPs.
A few places (Harlow, Basildon and Peterborough) have bucked the trend and increased the Conservative majority.
PS. My district has returned an independent residents' party for the third time running. We're still one of the safest Conservative seats in the country at national level.