Hi there,
Transitions to PE are quite common in London from MBB (especially Bain, given the private equity connections and strength of our Private Equity team.
If you have worked on Due Diligences, this is certainly extremely transferable as part of PE interviews involve a "case" fairly similar to a consulting case, but focused on either DDing a company, or finding value creation opportunities. You will also almost certainly be asked questions about previous DDs you have done - here be ready to go quite deep and answer both broad and deep questions. I'd suggest you sit down and reflect on each DD you did, and relook at the material - then think about how it links to the industry you worked in, the specific problem you looked at (e.g. how recession-resilient is this business?) to be able to engage in a discussion with your interviewer.
What you won't get from Consulting for interviews are mainly two things:
1) Deep knowledge of financials - e.g. deeper understanding of the links between the financial statements, importance of working capital, etc.
2) LBO/Valuation Modelling skills - you get these in consulting but mainly to make market models, not LBO models, so you need to practice this.
However, you should be able to pick this up fairly easily by dedicating some time to it. There are plenty of resources online - I would take a look at the Wallstreetoasis that has some good discussions and also (paid-for) materials on prepping.
Regarding Network, I would start reaching out to headhunters now. At least in London Network is not essential if you work at a top firm - most people just get invited through head hunters.
Good luck! PE transition is not easy for consultants, but it is certainly doable!