There is no legal requirement to adopt standing orders, so it's a matter of choice. LGA Schedule 12 para 42 states:
"Subject to the provisions of this Act, a local authority may make standing orders for the regulation of their proceedings and business and may vary or revoke any such orders."
Having said that, if you have adopted the NALC model document, para 5j(ix) requires the council to review and adopt appropriate standing orders and financial regulations at the annual meeting every year.
The model also includes at 18a:
"18. FINANCIAL CONTROLS AND PROCUREMENT
a. The Council shall consider and approve financial regulations drawn up by the Responsible Financial Officer..."
The implication in both cases is that it's a full council responsibility, but that is not stated specifically.