Well, Lange did not exactly cease to exist after World War II, the situation is far more peculiar - and one of the sources for certain rivalties between the two companies until today. In 1945, Glashuette's watch industry was heavily bombed, and what remained, was disassembled and confiscated by the Soviet occupation forces, including all company archives. the only archival substance left was that of Lange & Soehne, since that manufactury was hit so massively by the bombing attacks, that it was considered worthless by the Soviets. Thus, substantial parts of the Lange archives survived in German hands.
Later, the communist German govermnment nationalized ALL watch manufacturing companies in Glashuette, including all the different suppliers of balances, screws, springs, bridges, jewels, dials, hands etc, and concentrated them into one big company, the Glashuetter Uhren-Betrieb (GUB). So Lange & Soehne was part of that fusion.
After the downfall of the communist government in 1989, Walter Lange tried to re-establish his family's company in Glashuette. However, he had to accept the reality that "Lange & Soehne" still was but a part of the GUB. Therefore, he had to purchase the right to use the name "Lange & Soehne" from the GUB, whereas his efforts to take over the old Lange archives, too, failed.
The GUB was administrated by the so-called Treuhand, a 'trust', comprising all industries, which formerly were state-owned by the German Democratic Republic, which had become a part of (West) Germany in 1991. The Treuhand tried to sell the GUB in order to retain it as a working and healthy company. However, the first try proved disastrous, when the GUB was sold to the already weak France Ebauches, which hoped to use the cheap workforce to come on its feet again. IN spite of that, France Ebauches went bankrupt, and only political assistance prevented all the machinery and the archives from being sold. Then the Treuhand itself tried to keep up production, until a better deal could be found. During that time, rather cheap watches with own and Swiss movements were made with the "Glashuette" logo on the dial. Finally, the Bavarian businessman Heinz Pfeifer purchased the desolate company together with some investors, and what was believed impossible, was achieved by him within an incredibly short time: The GUB (with its new brandname, "Glashuette Original") succeeded in securing its place among the top ranking watch companies, without any assistance from Switzerland.