The main benefit from privatization was a significant reduction in fares combined with massive growth in availability (people take for granted things today that were huge problems then: punctuality, safety, etc.). The rail network in the UK is publicly-owned, what is private is the franchise to operate on a certain route. The lack of investment, if any exists, has come from the govt...and I am not sure there is a lack of investment either...HS2 is going to cost £100bn...£100bn is a large number (as ever, I think people mean...London and the area around me isn't getting enough, this is also false, London receives obscene amounts of investment relative to the rest of the UK).
Also, trains in Europe aren't really cheaper. Tickets are subsidised in Europe, in the UK passengers pay. The main constituency for subsidising the travel costs of wealthy commuters is...wealthy commuters. I am not sure why that idea is inherently attractive (and again, it is worth mentioning...our safety record is best in Europe, there is no real difference in punctuality, our fares are marginally higher but the ecosystem we have is far more robust).
TfL is a great example: sucks down tens of billions in public money to subsidise the wealthiest people in the country...the North-East is still missing motorways and we are spending £20bn on the Elizabeth line...okay. The cost of a fiber roll-out for the whole of the UK is less than the Elizabeth line. The problem is that this has no end because the more infrastructure you build, the more agglomeration, the more infrastructure...London had this problem in the late 19th century with slums, the solution wasn't more density (also, we had to bailout TfL twice I believe, they are not an example of the system working, tbf there isn't a lot management can do with the workforce).