Why an empire might use Singaporean or Qatari warship designs

Could a non-declining British Empire plausibly operate Singaporean- and Qatari-designed ships?

Yes; if naval architects and design institutions were based there, those regional designs would be entirely plausible.

Warships would naturally reflect the design centres available to the state. A surviving empire that educated and employed naval architects in Singapore or Qatar could plausibly assign complete warship projects to those institutions. The nationality attached to a design would therefore follow the location of skilled people and organizations rather than an immutable metropolitan rule. British wartime use of Australian designs provides the historical analogue for this more geographically distributed imperial design system.