The British were aware and unaware of Germany rearmament to varying extents depending on the service arm. I am also going to assume that when you say army you are referring to the Wehrmacht (Armed Forces) as a whole and not just the Heer (Army). The British in response to this had little political or economic capital to expend to stop German attempts to re-militarize due to the post-WWI situation in each country. It should also be noted that the rearmament started during the time of the Wiemar Republic and then expanded during the Nazi-era after Hitler took power.

The Luftwaffe (German Airforce) had to rearm in almost total initial secrecy as the Treaty of Versailles forbade Germany from having an airforce at all. The Germans established various Glider Schools and the German Air Transport schools to train pilots covertly under the guise of civilian and commercial aviation. This created a cadre of experienced aviators that the Luftwaffe could draw on for instructors and as their initial wave of pilots^1. As for the other service branches they had pools of experienced soldiers and sailors to draw upon that were demobilized in accordance with the treaty of Versailles that limited the size of both the Navy and the Army.

German rearmament only really started in earnest after the Nazi Party took power in 1933. Germany participated in the world disarmament conference of 1932 but refused compromises made by the British and French and thus the consensus there was that to bring Germany back into the negotiating table that a resolution giving equal armament rights would need to be reached. This would be the first clear indicator to the British and French that Germany would not comply with treaty law or efforts to prevent an arms race in Europe. On 16th March 1935 Hitler finally made the rearmament public by announcing German intentions to renounce the restrictions placed on them by the Treaty of Versailles and reinstating conscription. This was quickly followed up by the Anglo-German Naval Agreement in the same year which pegged German navy tonnage to be 35% of the Royal Navy and removed the Treaty of Versailles limitations that prevented the Kriegsmarine from exceeding the size of 6 Battleships and 12 Cruisers of both sizes. The Luftwaffe was also officially established in this time thus declaring that German had rearmed to a full tri-service military that was not bound by former restrictions.

Germany proceeded to then test the resolve of the Allied powers to stop German aggression. Hitler first ordered the reoccupation of Rhineland which was against the treaty of Versailles. Despite this blatant act of aggression the French and British did not respond by mobilizing their military for war to repulse the German occupation force as was the agreed arrangement. Instead they issued diplomatic objections which only emboldened Hitler to continue his rearmament. This was followed upped in 1938 with German union with Austria which was another violation of the treaty of Versailles as well as the demand from Sudetenland from Czechoslovakia which led to the British further appeasing Hitler in the Munich agreement. This finally culminated with the Danzig Ultimatum in 1939 which finally forced the hand of the Allied powers as they had guaranteed Polish independence. By this time the Allies were fully aware of German intentions due to Hitler's ultra-right facist rhetoric, his book Mien Kampf and escalating actions he had taken over the years. Thus, the British were fully aware of the German military's growing strength by at least 1935 when it was announced publicly by Hitler.

As to why the British did not attempt to stop the Germans through force of arms is largely due to the economic and political situation of the time. By the time the British were at least privy to the rearmament of German, they were already deeply affected by the Great Depression limiting their economic ability to sustain their own army. In addition there was a great anti-war sentiment among the populace with the Great War still in living memory thus creating political constraints for the government to take overt military actions. It is due to this that the British chose not intervene in the re-militarization of the Rhineland as they lacked the means to enforce it by themselves and the lack of backing from the French which were in a similar position mean that there was insufficient joint political will for it. There were calls for rearmament within the British government however there was a lack of political will to do this with the rearmament faction being considered warmongers or alarmist. The eventual British foreign relations strategy was largely helmed by Chamberlain who embarked on a policy of appeasement in the hope to placate Hitler's ultra-nationalist demands and eventually satisfy him.

They definitely knew that Hitler was potentially going to carry on his Lebensraum agenda, however they hoped they could avoid war when possible due to the high toll of the last one and their lack of economic ability to sustain one.They thus believed it was possible to placate him. While a flawed assumption today as we benefit from historical hindsight, it was not a totally unfeasible idea although an idealistic one. There was also a very high political cost to stop him earlier on in 1936 as his perceived threat level to the populace was not high and thus the government would receive little backing in their action.

Sources:

1. The Blonde Knight of Germany - Raymond Toliver

The Gathering Storm (The Second World War) - Winston Churchill

The Illusion of Peace, International Relations in Europe 1918 - 1933 - Sally Marks