My daughter who is a US Citizen resides in London and works in London. She is in the
Process of buying a flat in London.

Question #1 If I am a cosignor on the mortgage and she resides in the residence and
I make the payments, can I deduct it even if her name is the one the deed or whatever
The equivalent of a deed is in the UK. If the answer to this question is no then:

Question #2 If I actually hold the mortgage in its entirety and make all of the payments
Can I deduct the mortgage interest?

If the answer to both of these questions is no, how would I be able to structure it so I could
Take the deduction.

Thanks in advice for your answer(s)

This is from IRS Publication 17: Your Individual Income Tax
Generally, home mortgage interest is any interest you pay on a loan secured by your home (main home or a second home). The loan may be a mortgage to buy your home, a second mortgage, a line of credit, or a home equity loan.

You can deduct home mortgage interest only if you meet all the following conditions.
1. You must file Form 1040 and itemize deductions on Schedule A (Form 1040).
2. You must be legally liable for the loan. You cannot deduct payments you make for someone else if you are not legally liable to make them. Both you and the lender must intend that the loan be repaid. In addition, there must be a true debtor-creditor relationship between you and the lender.
3. The mortgage must be a secured debt on a qualified home. (Generally, your mortgage is a secured debt if you put your home up as collateral to protect the interests of the lender. The term "qualified home" means your main home or second home."
For details, see Publication 936.

So to deduct your mortgage interest, it must be your first home or second home, loan must be secured against the home, there must be true debtor creditor relationship, you must be legally liable for the loan, and must pay yourself.

One other thing - from Pub 963: http://www.irs.gov/pub/irs-pdf/p936.pdf

The mortgage must be a secured debt on a qualified home in which you have an ownership interest.

So if you are not the ower or co-owner, no mortgage interest can be deducted.