To be fair that isn't quite the case - Britain was rather dragged into a war not of its making and we did it to honour an absolutely aged treaty - The origins of the First World War are slightly more complex and Russia's early mobilization in defence of Serbia was closer to the actual spark of world war - to put it simply - ethnic Serbians stuck under austro hungarian control in Bosnia were desperate to be free in turn this lead "serb nationalists" to assasinate the Archduke Franz Ferdinand - Austria issued an ultimatum to independent Serbia that they would never have been able to accept - Austria was sure that Russia wouldn't mobilize in defence of Serbia if it came to war but to insure themselves Austria asked the Germans for guarantees that they would meet their treaty obligations if Russia did declare war.
Austria declared war on Serbia on July 28th 1914, Russia then announced mobilization (the pan slavic movement and the general popularity of the idea of Russia supporting the orthodox slavs living in the Balkans was deeply embedded and remained strong). Germany deeply militaristic viewed the long process of Russian Mobilization as a declaration of war on Austria Hungary and promptly declared war on Russia on the 1st August. France, bound by treaty to Russia, found itself at war against Germany and Austria-Hungary on the 3rd of August. Germany in accordance with plans laid down years below had already decided in the event of war against France invasion through Belgium (the quickest route to Paris) it was their invasion of neutral Belgium that caused Britain to entire the war. Under treaty Britain had a very loose but certainly moral reason to defend France - but her declaration of war on Germany on the 4th August was primarily prompted by german troops crossing into Belgium and the Kings appeal to Britain under a 70 odd year old treaty guaranteeing Belgium's independence. German war planning had always assumed that Britain wouldn't go to war for France or for Russia even if they did find themselves in a war on two fronts.
Russian action in 1914 did little to bog down the divisions in the west because of the variety of war planning conducted by all sides long before the conflict began. Germany wanted six weeks (the time it would take the Russians to mobilize) to knock the French out of the war and then concentrate their army on supporting Austria in the east. They very nearly achieved it and perhaps would have done if their top brass had been nearer their front, if communications had been better and they'd been better enabled to supply their troops at speed. The Allies (britain and France) were better able to supply their front lines.
Russia's war planning was less well conceived - and Grand Duke Nicholas had actually played no part in their creation which might explain why disaster at Tannenburg happened - the second incursion into East Prussia and Galcia was of course more successfull and by the end of 1914 Russia controlled much of Galicia forcing Germany to provide more troops to assist the Austrians. However Russia with the worlds largest army had a poorly equipped, poorly supplied one which was why she suffered such heavy losses. The German high commands decision to make its main focus the eastern front was just as much to do with the fact that their troops were bogged down in trenches in France with little chance of movement than the idea that Russia was sending wave after wave of men to the eastern front keeping the pressure on to aid her western allies
Nor in fairness can you blame the war entirely for the disaster that followed - Russia was in a state of semi permanent revolution or revolt since the disaster of the Japanese War and the aborted revolution of 1905. Strikes were rife in her industries in fact many members of Nicholas' government thought war might reaffirm the social order - it might have done had they had a decent infrastructure to enable them to supply the millions of men they were sending to die. Had Nicholas made his seperate peace with Germany it would have been as politically disastrous for him as carrying on - particularly after his decision to appoint himself commander in 1915 which made every defeat the personal responsibility of the Dynasty as many of his family had realised when they begged him not to do it. Its also debatable to use the arguement that had he made peace with Germany german divisions in Russia would have turned the tide on the eastern front - without Nicholas its highly likely that America might have joined the war earlier which would have made a significant difference.
Nicholas II lost his Empire and tragically his life for a great many reasons - the war might have speeded up the collapse - but his "loyalty" to the allied cause is a very small part of that.
Whilst i am aware that around 20,000 russians (Who later mutinied) fought in France (which given the size of the russian army wasn't that many at all) i don't have full figures so i won't argue that point.
To offer some one asylum is one thing - the other side have to let them go and despite kerensky's later comments by the end of march 1917 it was clear that it would have been difficult for the provisional government to have achieved that.