The best general figures I've seen for the Allied expeditionary forces at their strongest in the Crimean/Black Sea theater in 1854 are (roughly) 60,000 French and 27,000 British; source is "A Brief History of the Crimean War" by Alexis Troubetzkoy, Carrol & Graf, 2006, p. 169.
Obviously, the total number of Allied troops (including, of course the Turks and Sardinians), deployed to and present for duty in the Crimean over the next 24 months varied greatly, but it gives a decent snapshot of what each of the dominant Western powers could assemble for an overseas expedition against a comparable power at this time.
So the question is, absent the French Army (say that N. III is willing to support the Turks with naval forces, but not an expeditionary force; he has too many issues in Germany and Italy to worry about), do the British:
1) Mount a Crimean expedition with their own troops, plus whatever Turkish forces can be sustained, and try and recruit additional allies (Sardinians, the historical German, Swiss, and Italian "British" legions, etc?)
2) Mount an expeditionary force somewhere other than the Crimea, leaving it to the Turks to defend their frontiers in the Balkans and/or Anatolia without an "Allied" diversion?
3) Simply blockade what Russian ports they can?
4) Give the whole thing up as a bad job and insist the Turks accept the provisions of the 1854 Vienna agreement?
5) Say, "righto, let's solve that Eastern Question now, rather than in 1919," and join in with the Russians?
Best,