I'm too old for the sponsorship schemes. However I did have a conversation recently with the head of one of the FTO's who runs a couple of the selection procedures for the airlines here in the UK, and he said that the degree did have a bearing. However, unfortunately for you, it was more the subject than where you did it. It was idle chit-chat, but he stated that preference was given to those with an aeronautical-related degree. The reason for this was it gave a far higher chance of the person completing the ground school with first time passes. He also looked for Physics and Sciences at A-levels and GCSEs.
He did not mention the degree result; if a first is looked on more favorably than a 2.1 etc. So if you have a first in physco-babble from Cambridge, I don't think it really matters. In summary, we’re not talking about the same criteria some of your colleagues would have been subjected to when applying for the FCO or certain banks in the square mile.
Personally, I would look long and hard at Flybe before I committed the emotional cost of going for their scheme without the requisite science and aero background. Of course, this was only one scheme, and not representative of them all; I get the impression that British Airways, on their old cadet scheme, did not weed out candidates in this manner.
The nature and style of your question raises some further points you must consider. If you won't apply to sponsorship schemes because too many others apply, then I would say that you need to examine your motivation for the career. Secondly, if you are only looking at doing the career if you can get sponsorship then, again, you need to look at why you are doing it. Sponsorship schemes are highly over-subscribed, and your motivation MAY be looked at closely.
As for this site PPRUNE, well some refuse to look at it, other say they don't, but secretly look at it, and there are certain ops rooms that print topics out and leave them around for people to browse. Generally it is a good source of information for the wannabe. Reading the message at the bottom of all forums will tell you that there are occasions where people hang out their dirty laundry and there are others who have hidden agendas.