I'm a resident of the ward in question, and delighted by the result. If anyone is interested, this is what I think happened. If not, ignore!
Whatever Sarah Brown is saying now, the Lib Dems seemed to think they would win this, and they are not happy with the result. Brown will doubtless be back to contest the next election in the ward next May (no-one else in the Local Lib Dems seems to have a shot at standing as a candidate in the ward). The Lib Dems ran a nasty campaign insinuating that Ann Sinnott and the rest of the Labour Party are bigots (this from the party led by Tim Farron until not long ago), tried to take credit for a local road safety initiative that Labour had actually pushed through, and tried to sell Brown as someone who had been a great councillor in the past. Because Sinnott was handing out WPUK leaflets, the LD LGBTQ+ twitter account was spreading a load of (formaldehyde-pickled) bollocks about transphobic hate literature flooding the ward in the run-up to the election. They also tried to pitch this as an election about Brexit (the ward voted Remain by over 80% in 2016). None of that did any good.
They did however, cut the Labour majority significantly, and are trying to spin this as some kind of turning of the tide towards them. This is almost certainly wrong. As some people have pointed out, one very likely reason that the Labour and Green vote went down and the Lib Dem vote went up was that the election was held in the university vacation. There are a lot of Cambridge University and ARU students in the ward, and it's likely that a lot of those who bother to vote will be voting Labour or Green. Brown picked up more votes this time, so some people have switched, but the number of missing Labour and Green votes this week compared with the vote in May is much greater than this couple of hundred extra votes. The Lib Dems won't be so lucky next time round, since council elections are normally held in term.
I had quite a lot of firsthand experience of both Sinnott and Brown as councillors and they were both pretty useless at engaging with residents and issues in the ward. I don't know anyone in the ward who found Sinnott to be remotely engaged or receptive as a councillor. At area committees (when residents turn up to raise issues in an open forum), Sinnott was prone to telling people concerned about things like road safety that these were first world problems and they shouldn't be complaining because lots of women have to live with domestic violence. I don't know any ward resident who ever got a substantive reply to an email they sent her about any council business. It's really not the case that Labour has lost votes because a much-loved councillor has stepped down. She has been an excellent campaigner for women's rights inside and outside the council, and I greatly respect her for that, but she wasn't at all suited to being a councillor dealing with residents' issues.
Sarah Brown was just as disengaged, it always seemed to me, and seemed to spend almost all of any meeting I was at glued to her ipad, rather than engaging with the tedious business of local democracy. I don't know of anyone who looks back fondly to the golden era of having Brown as a councillor, whatever the Lib dems are trying to claim. That's particularly true for anyone who has ever encountered Brown on social media - she's very aggressive, and not someone who strikes you as able to engage with any interest or sensitivity in local problems. (This is completely different from her partner Zoe O'Connell who seems to be a good ward councillor.)
So, I really don't think the election and its result was about the bigger issues, however much it looked as if it could be. Brown probably wanted to make it all about horrid Labour transphobes, but I doubt that did very well on the doorstep (that aspect of the campaign seemed to get toned down quite quickly). In the end, the permament residents round here are more likely to vote on the basis of things like residents' parking than whether they buy into Brown's hyper-aggressive brand of TRA.