Where can I apply, for my Travel Authorisation to USA?.?

Do you mean ESTA? See site below.

Go here:
https://esta.cbp.dhs.gov/esta/esta.html?_flowExecutionKey=_c832EE0D9-1DAA-32C7-2534-BAD482E6E6A4_k7A5F15A4-3777-6179-98EE-5DB234397272

Also check out the information here:
http://www.britishairways.com/travel/usawaiver/public/en_gb

For visa related questions, you would speak to the embassy concerned and details can be found on the UK Foreign Office website.

I have provided links to the FCO site, the US embassy website, and the website you can apply for your Travel Authorisation.

This is the website you need:

https://esta.cbp.dhs.gov/esta/esta.html?_flowExecutionKey=_c2FA8DFB1-E0F5-C7DF-7D24-185A81194334_k6A32F5B4-A24F-D199-D995-8D82C5C8F927

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