Just as a matter of interest, how long does it take after applying to actually receive US social security?

It's your choice; you specify what month you want it to start. Then, if I remember correctly, they pay it to you a month in arrears.

How long does it take to begin to receive it, though, from the time of applying?

I did the online application just before just before Christmas 2014, the 1st payment I got was paid into my UK account on 3rd March 2015, it included the payment for February and the arrears due. It is now paid monthly in arrears on or around 3rd of each month.

Hope this helps