How can I pay for flight school in the UK?
There are specialist firms providing loans for aviation training, however it is unlikely that you will be able to get a loan of the amount you would need for an integrated course without some form of security.  

The security usually takes one of two forms.  The first is a charge on a property (belonging to you or your parents).  The other is an airline agreeing to act as a guarantor/sponsor, however this doesn't appear to apply in this case as this particular scheme states it is fully self-funded.

If you cannot raise the financing for this particular scheme, you can apply to other schemes that provide some sponsorship or underwrite the loan as they become available.  Naturally there tends to be more competition for these schemes.  Some are targeted towards under-represented potential pilots, such as the Amy Johnson initiative by Easyjet, where they will guarantee/underwrite the loan for a limited number of female trainees.

It is also possible to undertake modular training, where the cost can be closer to £50,000.  There are a number of organisations that can provide funding towards some of the costs, particularly for young pilots.  These include The Honourable Company of Air Pilots, The Air League and The Royal Aeroclub Trust.

Before spending significant sums on training, it is advisable to get a class one medical, to avoid possible disappointment after spending significant sums of money that cannot be recovered.