It seems strange to me that a European neighbour would be able to amass an army of such size without provoking an arms race.
There are a wide variety of reasons why Britain maintained a small, professionalized army in the years leading up to the First World War and I won't try to address them all. I'll try to hit some of the main points and other contributors can free feel to pop in with anything I've missed or neglected.
Firstly, the senior service of the British military was, since the time of Drake, the Royal Navy. The BEF was designed to be ferried around the globe by the Royal Navy to deal with colonial threats. The Navy, rather than the Army, was seen as the protective shield of the British isles. The Army was principally a force meant to deal with colonial trouble in hotspots around the globe.
Secondly, the shifting system of alliances throughout the 19th century placed greater emphasis on the Royal Navy than on the BEF. For example, before the Entente Cordiale settled many of the colonial differences between Britain and France in 1904, France was viewed as one of, if not the primary, threat to British dominance overseas. Defeating France primarily meant defeating the French Navy rather than having to destroy France's continental forces. During this period, Britain could enlist the continental military strength of another Great Power, Germany, to rival that of France. After the Entente Cordiale, Britain could rely on the French Army to provide the vast majority of continental forces to oppose Germany, which is exactly what happened during the Great War, at least prior to 1916 when Kitchener's New Armies began to make their appearance on the Western Front.
Thirdly, British foreign policy as regarded the continent tended to focus on supporting several smaller allies against whoever the major European power happened to be at the time. During the Napoleonic Wars, Britain supported Russia, Prussia, Portugal and a whole host of different allies against Napoleon and his client states. By supporting various smaller allies against the dominant land power (whoever that might be) Britain maintained a continental balance of power which allowed Britain to intervene decisively to tip the balance of power in whichever way favored British policy.
Lastly, As /u/histomat stated, Britain did see the preparation of the continental powers in increasing the size of their armies. In 1907, Parliament established the Territorial Army which consisted of 14 infantry divisions and a similar number of mounted brigades. Each Territorial battalion was paired with a specific "regular" Army regiment so that in war time the standing strength of a given "regular" Army regiment could be bolstered by the addition of Territorial battalions.
This type of situation had already been instituted in France and Germany, although both of those countries required military service for a designated period by men of certain ages. For example, an 18-year-old might be required to serve 2 years of military service and then was shifted to a reserve force in which he trained on a monthly or semi-annual basis until he was 30 years old. After he turned 30 he would be shifted to a second tier reserve where he no longer trained, but remained eligible to be called for war service until he was 50 years of age. So while standing armies of the continental powers were quite large they also possessed reserve forces similar to the British Territorial Army that could be mobilized during time of war.
It's also important to note that arms races were a major part of the pre-World War One era. France, Germany and Russia all engaged in efforts to create larger and large armies, while Germany and Britain engaged in a supremely expensive naval arms race.
A great read on the naval arms race is
Robert K. Massie, Dreadnought: Britain, Germany and the Coming of the Great War, 1992.
For more information on the BEF, I suggest:
Richard Holmes, Tommy: The British Soldier on the Western Front, 2005.
For more on British foreign policy vis a vis balances of power in the pre-WWI era,
Gary Sheffield, Forgotten Victory: The First World War - Myths and Realities, 2002
has a succint summary of British efforts to maintain balances of power on the continent over the 19th century.
They certainly saw it as a threat, there were several fiction books written predicting an invasion. First, for a long time the British hoped to remain neutral and to stay off the continent. Second, Britain was at risk of a revolution from communists and at risk of civil war in Ireland. They had to consider how building their forces would look to their citizens and if the lower classes would even fight. Also how would building their army be viewed by other powers. Third, even with Germany's massive effort to build their navy the British still greatly out numbered them in dreadnoughts and the navy was seen as Britain's safety, not the army. Last, after the problems in north Africa and the Balkans were resolved there was hope that diplomatic measures could truly resolve most issues. I also wonder if family issues were at play. I mean the Kaiser, King of England, and Tsar were all close relatives.
Sources: Margaret MacMillan's The War That Ended Peace and Max Hastings' Catastrophe 1914.
I'd be interested in a source on the Imperial German Army being 2 million men prior to mobilization.
This suggests 840,000 in July 1914: