Alternator kicking in and cutting out 2004 Cooper

Hope someone can help me, my until now totally reliable 2004 Coooper has developed a charging problem, here's the story bulleted to make it easy :-

1. Went to start car, completely dead, not even enough life to try and turn the engine (it was a really cold day so assumed battery had died).
2. Removed battery and put it on an Optimate overnight, Optimate got it back to registering all fine.
3. Reinstalled battery, started car and it worked perfectly for 3 weeks no issues.
4. Got caught in a traffic jam immediately after starting the car, after about 10 minutes noticed the battery light come on, stuck in the jam the battery light would flicker on and off, then the power steering shut down, after about 20minutes the traffic cleared and I got going, power steering returned but the battery light kept flickering on and off, car felt like it would die but didn't and got me 15miles home.
5. Went and bought a new battery, installed it, car started fine but the battery light flickered again.
6. Got out a multimeter, across the terminals the voltage fluctuated between 12.5v and then jumped to 13.5v and would repeat, about 20 seconds low then 20 seconds high.
7. Thought alternator fault, next door is a car dealer he took the reg and he kindly sourced me a new Delphie alternator at trade price, I dropped it and car off at local garage he suggests, they fit it,.. and the same fault, no change - they suggest power steering pump as they felt the battery light came on with movement of the steering wheel (I'd not noticed that).
8. They get an automotive electrician in to check, he can't pinpoint anything for sure causing the fluctuating voltage and suggests its a duff alternator or control module in the clocks?, doesn't think its the power steering pump as disconnect it completely and the same problem persists, I ask him to check body control module in the foot well see if its wet, it isn't it looks dusty looks like its never seen water.
9. I get car back I check every earth I can find, remove, wire brush any corrosion off put back on, car idles fine for about 45seconds then the damn battery light flashes again. I can hear the alternator kicking in and then shutting off, I plug in a scan gauge it can't find any fault codes but shows voltage holding 12.5v then jumping to 13.5v then dropping etc.


Does anyone know what I should look at next? Reading through the forum it seems some people experience all sorts of electrical problems with the route cause being the power steering pump, is it likely that even though I've been told problem still happens with it unplugged, does it have any roll is regulating voltage?

thanks in advance for any help, the BMW / Mini dealer near me says 3 weeks until they'd have time to look at the car.. :-(