The supersonic P.1154 and Britain's carrier choice

Did Britain consider a supersonic Harrier-like aircraft for its carriers?

Yes. The P.1154 concept depended on retaining larger Hermes-sized carriers; the move to Invincible-sized ships weakened its prospects.

Britain examined the P.1154, a supersonic carrier-capable VTOL design with demands beyond a faster Harrier. Its prospects depended on the ships the Royal Navy expected to operate. Continued construction of Hermes-sized carriers might have justified development, whereas the move toward the smaller Invincible class weakened the case. The Sea Harrier emerged as an adaptation to the carrier force Britain actually retained.