MBB consultant targeting mid-market PE in Milan / London — how does the headhunter process actually work here?

Current MBB consultant, ~2.5 years in, based in MIlan. Generalist background but most of my work has been transaction-led: buy-side commercial and industrial DD, a corporate carve-out I covered from diligence through signing, closing and integration, plus post-acquisition value creation work for a sponsor-owned business.

Looking to move into mid-market PE, ideally Milan or London. A few questions for people who've actually been through it:

1. Milan — which headhunters genuinely cover Italian mid-market PE at the associate level? Most of the threads here are London or US-focused and I'm struggling to find anything reliable on Italy.

2. London — I know the main names (KEA, Blackwood, Dartmouth, PER, Walker Hamill, Nordbridge). Which of these are actually worth the time for a consultant rather than a banker? My understanding is Walker Hamill leans consultant-focused but I'd like to hear from people who've worked with them.

3. Cold outreach — realistically, do you get anywhere emailing headhunters without a referral, or is a warm intro effectively mandatory? 

4. Timing — is there a meaningful season for European mid-market, or is it genuinely rolling?

5. Modelling — for consultant candidates, how hard do funds actually go on the LBO test at the first technical round? Trying to work out how much runway I need.

Happy to pay it forward once I'm through this. Thanks.