The British expeditionary force or the BEF became involved in the war because of Belgium. The Germans decided to go through Belgium as part of the Schlieffen plan, and though they knew that the British might become involved they didn't think that they would and they also felt they didn't need to fear the tiny British force.
The Schlieffen plan expected the Belgians to give in when the Germans marched through the country. However they instead fought the Germans and though they wee massively outnumbered the resistance brought time for the British and French to get their troops in to the war to help Belgium. The French were fighting the war on another front so the main force with the Belgium's was British. This force can quite possibly be said to be key to the failure of the Schlieffen plan as although they were hugely outnumbered they combined with the Belgium's slowed the Germans down enough that the 6-week period the Germans had allocated to take France was impossible to achieve.
They slowed the Germans by organised retreating which meant he Germans could only advance slowly.
This may be considered a major factor but it is not the only thing that should be considered concerning the failure of the Schlieffen. Another key factor would be Russia. Germany's reason for the Schlieffen plan was that they did not want to be fighting a war on two fronts, the French and the Russians. They believed that it would take Russia 6 weeks to mobilise it's army and sop if France could be taken in less than 6 weeks they would be able to get back to the other side of Germany to fight the Russian army. They intended to take Paris and so force the French to surrender. The problem occurred when not only did they fail to take Paris and France as quickly as they needed but Russia was able to mobilise it's troop's inn about 2 weeks far quicker than expected. This left the Germans open to attack from a big force from Russia and forced the Germans to send some troops to reinforce the army on their border with Russia. This weakened their main force and meant the Germans had to constantly be thinking about the Russian threat all through the war, and kept the German forces strained by fighting on two fronts.
Another cause of the failure of the Schlieffen plan was that the Germans had very little sport. Their main ally Austria-Hungary was more concerned with Serbia than with the rest of the war which meant hat the Germans had to fight almost on their own stretching troops to the limit. The plan relied on the fact that the German troops were the best in the world and that they had the biggest army in the world but they were forced to call up millions of reservist as they would otherwise not necessarily of had enough troops.
Another major issue was the Schlieffen plan it self. This plan had a lot of risk in it as it relied on a lot of belief like the British not joining the war and the Russians not being able to mobilise the army quickly. It also had faults to do with the running of the plan. When the Germans pushed further in to France their supply trains for food and other materials began to be very stretched and this left he army without crucial supplies sometimes. The plan also thought that the French would push to try to get back the provinces of Alsace and Lorraine. Though the French did indeed attempt to regain their lost provinces the Germans had thought that France would be occupied solely by the objective of getting the two provinces back. However the French instead realised the danger to Paris and sent their army back to Paris by train to stop the threat. This not only help foiled the Schlieffen plan as it made it very hard to get Paris, but added more strength to the British and Belgian troops fighting the Germans.
After considering all the factors and reaching a conclusion, I would say that the British expeditionary force played an important role that was crucial to the failure of the Schlieffen plan. This crucial part is however because of the Belgian effort combined with the BEF, and it is unlikely that the BEF could have made as much ad difference to the war on their own.