Did British enter World War 1 when the German troops crossed into Belgium?

Yes. First the British called on the Germans to withdraw from Belgium, and set a deadline. When the Germans did not withdraw the British entered the war.

Britain and Germany (actually Prussia, of which Germany was a successor state) were two of four Great Powers pledged in an 1839 Treaty to perpetually respect and defend Belgian neutrality. German war plans for a generation had cynically disregarded this Treaty obligation in their scheme to outflank French border defenses by going through Belgium. When Britain raised the issue of the Treaty in demanding German withdrawal from Belgium, the German Foreign Minister denounced the Treaty as " a scrap of paper".