I'm not sure when the Second Opium War broke out, but if you could somehow delay the Second Opium War until the same year as the Indian First War of Independence, then Britain is forced to expend her resources fighting both the Indian revolutionaries and the Qing Chinese forces. Although the British may win in China, she'd be exahusted fighting the Indian revolutionaries.
Two problems with this. One, it already happened: as Lord Elgin wrote in his diary:
5 June [1857]. I received letters from Canning, imploring me to send troops to him from the number destined to China... I did what I could, and in concert with the general [Ashburnham] have sent instructions far and wide to turn the transports back, and give Canning the benefit of the troops for the moment.
Two, the commitment of troops when the British did end up fighting (1860) was not exactly large- c.14,000, which due to an administrative error was larger than the 10,000 they'd agreed with the French.
General Commanding: Lieutenant General Sir James Hope Grant
1st Division (Major General Sir John Michel)
-1st Brigade (Brigadier Charles Staveley)
--2/1st (Royal) Regiment
--31st (Huntingdonshire) Regiment
--Ludhiana Sikhs
-2nd Brigade (Brigadier Sutton)
--1/2nd (Queen's Royal) Regiment
--2/60th (King's Royal Rifle Corps) Regiment
--15th Regiment of Bengal Native Infantry
-1st Battery, 4th Brigade Royal Artillery (Barry's)
-4th Battery, 13th Brigade Royal Artillery (Desborough's)
-No. 10 Company, Royal Engineers (Fisher's)
-Half of no. 8 Company, Royal Engineers
2nd Division (Major General Sir Robert Napier)
-3rd Brigade (Brigadier Jephson)
--1/3rd (East Kent) Regiment
--44th (East Essex) Regiment
--8th Regiment of Bengal Native Infantry
-4th Brigade (Brigadier Reeves)
--67th (South Hampshire) Regiment
--99th (Lanarkshire) Regiment
--19th Regiment of Bengal Native Infantry
-3rd Battery, 13th Brigade Royal Artillery (Govan's)
-7th Battery, 4th Brigade Royal Artillery (Sterling's/Moubray's)
-No. 23 Company Royal Engineers (Graham's)
Cavalry (Brigadier Pattle)
-1st (King's) Dragoon Guards
-11th Regiment of Bengal Cavalry
-19th Regiment of Bengal Cavalry
-2nd Battery, 4th Brigade Royal Artillery (Milward's)
Reserve
-4th Battery, 2nd Brigade Royal Artillery (Rotton's)
-Siege train and heavy artillery
-Madras Sappers
-1st Battalion, Military Train
To put it in perspective, there were 66 British and 12 Indian Army regiments in India in 1859, with 36 regular and 39 embodied militia regiments at home and 29 in the colonies (i.e., excluding India). In 1856, there had been nine regular and 113 embodied militia regiments at home, 25 regular regiments and 10 embodied militia in the colonies. Long story short, Britain wasn't at the end of its rope.