The United Kingdom needs more private universities. This would help specific institutions, such as churches and organizations to educate people based on their own respective values. Public education is a necessity, but private education gives students a special treatment that helps mold them into a more specific belief system. These universities exist in high numbers in America, and they would be good in the UK too.
Private universities always cost more money than public schools and this hurts students who may want to go to the school. Prices and tuition fees for private schools are a lot and the requirements are not always easy for students to meet in order to attend that particular school, that could be a problem when reflecting attendance.
As an American, I do not see the benefit for the United Kingdom to have more private universities. We have them here, places like DeVry University or University of Phoenix. They are basically nothing more than diploma mills that put students into debt, deliver very few skills, and are not taken seriously by hiring companies.
While the United Kingdom only has two private universities I do not believe this is an indication that they need more. In the United States private universities are often seen as overpriced options that sometimes do not offer comparable educations. The United Kingdom has many government funded universities that prepare many students every year and that should not be looked down upon.
The UK needs more public universities that are either completely free or mostly paid for by grants and loans. Same as the USA needs it. The cost of living in the UK is very high, and with so many students abroad wanting to study there, private universities wouldn't be a good idea.