Well, more than once the Brits had been actually encouraging the Russian naval presence on the Med. The 1st time it was 1st Archipelago Expedition during the 1st Ottoman War of CII, then Mediterranean Expedition during the War of the 2nd Coalition, 2nd Archipelago expedition (3rd and 4th coalitions) and the Greek War of Independence.
With the exception of Mediterranean Expedition, the rest had been done by the Baltic fleet, which obviously required the British cooperation. The relations were so good that when Russia changed sides after the 4th Coalition there was a Lisbon Incident: commander of the Russian squadron refused to obey the imperial order to cooperate with the French against the Brits and then, when the Brits blocked Lisbon, there were negotiations as a result of which the Russian squadron was to be escorted by the Royal Navy to
London without lowering the Russian flags and with the Russian admiral in command of the joint Anglo-Russian fleet.
So all that Mediterranean brouhaha is date by the later time when the Russian-British relations soured due to the Great Game. Basically, well before the CW the Russia was interested in the Ottomans
not letting the foreign warships through the Straits, not in getting into the Med, because their navy became obsolete and could not compete with the Brits or French. With a continued Great Game the British attitude escalated into a full-scale paranoia: during the war of 1877-78 the British politicians had been seriously talking about penetration of the Russian Black Sea fleet into the Eastern Med (presumably) knowing that Russian Black Sea Fleet did not exist (there were few ships of a coastal defense including the round ironclad and few armed merchant steamships, construction of the fleet started only in 1880s).
What would Russia do with Constantinople? On a positive side -
control access into the Black Sea. On a negative side - have countless problems with feeding its population. It can be
imagined (if one has a well-developed imagination) that at some unidentified point in the future the Russian Black Sea Fleet (which has to be built) would be sailing on the Med. But how and why would it be able to compete with the Brits I can’t imagine. The same goes for the merchant fleet: in OTL it was under developed and majority of the traffic had been done by the foreign ships (mostly British).