"In the UK you generally have a Master's already before beginning a PhD" That's not my experience - you can generally go straight from an undergraduate degree

"In the UK you generally have a Master's already before beginning a PhD" That's not my experience - you can generally go straight from an undergraduate degree to doing a PhD - I did and most of the people I knew did the same (and the people who had Masters either had done MEng as a first degree or had MSc conversion courses). NB There is also the weird thing where some UK institutions give you a MA for still existing after a few years....