In The Ascent of Money: A Financial History of the World by Niall Ferguson, he discusses (among many other things) the history of debt instruments. The truth is, the buying and selling of debt obligations has been around almost as long as debt obligations themselves.

Specifically, you may be referring to the changing of hands of the servicing rights of the mortgage. I have to delve a bit into current events here, but when you get a mortgage today (and this has largely remained unchanged since the sub-prime crisis of a few years ago), the mortgage itself is likely to be sold only once and only once - probably to Fannie Mae, Freddie Mac, or Ginnie Mae, depending on the type of loan, but sometimes to private companies (although not nearly as much since the real estate bubble popped). Then, the cashflow from hundreds or thousands of loans is sold to investors in the form of bonds. The company that actually accepts your monthly payment (say, Wells Fargo or Bank of America) does not own the loan. They accept a servicing fee each month to pay for their expenses in servicing the loan (I've seen between 0% and 1.25% of outstanding principal as a servicing fee), and pass the rest of the payment onto the trustee which administers the bond. It is not uncommon for the servicing rights to change hands during the life of the loan, and was quite common during the real estate boom in the US in the previous decade.

Away from current events now, and back to history.

Modern securitization as described above is a system that came about with the chartering of Freddie Mac in 1970, and the creation of the first residential mortgage backed security (RMBS) by the US Dept of Housing and Urban Development. But securitization of mortgages had been around since the mid 1800s. It's easier for a small local bank to loan out money when they can sell their mortgages immediately on the secondary market, rather than have their money tied up for many years. However, and getting to your question, even when the mortgage itself was sold, the mortgagor (usually a farmer) still made payments to the same bank that originally made the loan. It was not until the Great Depression and the creation of Fannie Mae that the secondary market included government sponsored enterprises (GSE). Before that, it was large, usually East Coast, banks that bought mortgages on the secondary market.

So I guess the answer to your question would be around 1970, with the creation of the first government sponsored RMBS, and then the late 80s, with the creation of the first private label RMBS. The RMBS structure allows Bank A to make a loan, Bank B to buy and repackage the loan, and Bank C to service the loan, potentially removing the originating bank (Bank A) from the equation before the first payment is even made (in all three cases, I am using the term "bank" loosely, because these are very different kinds of banks). In previous forms of securitization (the ones going back to the 1800s), the originating bank was either the owner of the mortgage, or at least retained the servicing after they sold it (otherwise, they wouldn't have gone to the trouble of originating it).

Besides the above book, my source for the second paragraph, where I discuss modern mortgage servicing, is that I've been in the industry for 10 years, including some very interesting front row seats working as an RMBS securities analyst from 2007-2009. I hope that's ok, and that I answered your question.