In the event of a company's bankruptcy, how does a SPV continue operations if it can no longer collect payments on collateral
Servicing and origination don't have to be done by the same firm. It is quite common that the financing company contracts out the servicing rights on the loans they originate. 
Servicers, like many other kinds of firms, can continue to operate after bankruptcy. Bankruptcy is what happens when a firm can't pay its bills which includes debt, it doesn't mean that the firm needs to be liquidated. For example, existing debt holders are not getting paid and sue to put the firm into bankruptcy. The equity holders are wiped out, the debt holders become the new equity holders, and with the reduced debt costs the firm can often cover its costs and be profitable (even though the debt holders took a loss on their investment). 
Servicing can be reassigned after servicer bankruptcy. In fact, not only does this happen but it is common for a servicer to change when a loan is sold. So transfer of servicing rights is commonplace.