Ok, here is what you will need to do.
First, if your cousin buys the rifle for you, it is called a 'straw purchase'. If caught, both of you will become guest of a federal prison.
If the firearm is to be exported out of the USA, it must be done via a munitions broker. You will need to apply for a US Export permit via the broker (as they must be licensed to export munitions). You will also need to complete an end-user certificate, purchase order, DSP-83, BIS-711 and pay around $500 to get the export licence. You will need to supply proof that you have an import permit or a waiver (on official government letterhead) stating that a permit is not required. This will all go to the broker to process with the US State Department. This will take time (months).
Once you have the Export permit, the broker can then go ahead with the purchase (they may or may not allow you to have the firearm supplied by a third party). The broker will then ship the firearm (usually by secure air freight) to the UK. The shipment is normally free on board, so you will need to get to the freight forwarders to pick it up (and get FAC signed, pay duties and VAT).
You won't be able to just take it out of the country. If you try and get caught, the US take a very dim view of it -
http://edition.cnn.com/2011/CRIME/01/25 ... a.guns.uk/
I went through most of this process and in the end pulled out of it as the cost of permits and shipping made it uneconomical to do it for a one off.
Regards,
MB