If a new airport by the Thames is built, then this is how it will be funded (if I was deciding this):
A new arms length-state owned company (London Airport Limited) would buy Heathrow, Stansted, Gatwick, City, Southend, Luton, Southend, Manson, Biggin Hill, Ashford and Cambridge Airports (and nearby hotels, offices and other airport related facilities)
It will issue long term bonds to the general public and other institutions (at a rate of lets say 4%) that will fund 60% of the overall cost, the rest coming from government
When this new airport opens, the other airports will either shut down or become general aviation airports (which can be done in a way that Biggin Hill has already done) and thus the land can be sold (and thus the revenue from the sales can go towards funding the project
As for if airlines will move to this airport, well since there will no longer be any Heathrow, Gatwick Stansted etc, then they will have to use this airport if they want to serve London, also the fees for using this new airport will be the same as Heathrow (or even less to attract more traffic)
As for if it will be profitable, well if we were looking for the most profitable solution, then we would be expanding our airports ASAP, but we cannot, because of all the political related s**t and the fact there has been no proper planning and co-ordination for London (and the South East) regions aviation infrastructure, in other words, if you want to see some returns on a new Thames Airport project, it will be in the long-term
Plus a new airport will be a massive boost for the economy, since it will create jobs and improve our airport capacity (however this project can only be done if the government cannot have the balls to approve a 3rd, 4th and maybe even 5th and 6th runway at LHR)