It seems to me that the privacy concern could be addressed by allowing people keep control of their own records.
To do this, instead of having a single repository of data, you would need a uniformly applied standard for submitting financial data.
I have a picture in my head of a uniform 'fact find' type sheet. The form would not have to be identical in its printed form for every application - it would just have to be uniform in the way questions are phrased and in the way that the answer data fields are named. For example, you couldn't have one insurance proposal asking if you had any claim in the past 4 years, and a different one asking if you had any claim inthe past 3 years. (well, you could, but it would add needless complexity)
If the 'fact find' sheet was uniform in its questions and in naming its data fields, this could then be used as the data source for filling in forms from financial institutions. For any other form, you could simply download the form and have a 'one click' filling in of it, in much the same was as you can have an 'autofill' for name and address details . A third party such as Google, (or perhaps the Financial Regulator...just joking) would have to provide a template where you could initially enter all of your information - and this template would have to have all the questions, and corresponding data fields, that could reasonably be requested by any financial instution.
I would be kinda surprised if this sort of thing was not already available. If it isn't, there is clearly a market opportunity.