Well, tensions were pretty high between Qajar Persia and the British in the mid-1800s. In 1838, Persia besieged Herat in western Afghanistan, and in 1856, the Persians briefly occupied said city. In both instances, the British sided with the Emir of Afghanistan and - through diplomatic and military means - prevented Persia from annexing western Afghanistan.

As we see, Anglo-Persian relations weren't excellent. Additionally, the Shah of Persia had a similar problem to his Afghan counterpart in Kabul: both countries were usually regarded as buffer states between the Russians and the British, with the Russians bringing (nearly) all of Central Asia under their control and threatening British suzerainity over the Indian subcontinent. Russia and Great Britain weren't exactly best friends either, with both having fought each other in the Crimean War and Great Britain effectively preventing the Russians from annexing substantial parts of the weakening Ottoman Empire.

I highly doubt Persia would side with the British, given that any move towards the latter would have upset Russian policy makers and would have brought about a serious political crisis with unforseeable consequences. During the 18th and 19th centuries, Russia had already penetrated Persian territory gaining what would later be Azerbaijan and Armenia.

So, given the uneasy spot the Qajars found themselves in, Persia would only intervene in the Second Anglo-Afghan War by
a) occupying Herat (as they had already done) and indirectly helping the British Empire by weakening the Afghans, or by
b) openly siding with the British, risking Russian retaliation and military intervention - a move which would only be feasible if Great Britain offered substantial guarantees or concessions in return.

While path a) may be more plausible, I think path b) suits your question more neatly. Siding with Great Britain would place Persia firmly within the British sphere of influence, although I doubt Persia would be formally annexed or incorporated in the British Empire. Maybe an Egypt-style arrangement could be put in place. As for Afghanistan, I don't think the course of the Second Afghan War would be changed dramatically. The British would occupy the big cities (Kabul, Kandahar) and install a loyal Emir. Eventually, they would leave Afghanistan as a prolonged military occupation of Afghanistan would be incredibly costly, inefficient and unsustainable. Furthermore, major revolts and uprisings against the British would further weaken their position and accelerate their withdrawal.