You are correct that Namibia was German Southwest Africa, until 1915. During World War I, there were several campaigns by French, British, and Belgian colonial forces to capture the German colonies of Togo, Kamerun, German Southwest Africa (namibia) Ruanda-Urundi (Rwanda and Burundi) and German East Africa (Tanzania minus Zanzibar).

40,000 white South African troops (as well as many more Black and Indian laborers who built railways, and other transport and supply roles) were involved in the campaign in Southwest Africa, and it was the South African general Lois Botha that accepted the surrender of German forces in may of 1915.

Namibia would continue as an allied occupied territory throughout the remainder of World War 1, but the League of Nations would appoint the Union of South Africa to administer Namibia as a mandate territory in 1922. South Africa continued to administer this territory throughout the second World War, and despite progressively louder UN condemnation and SWAPO/Namibian liberation struggle, until 1990. So, to answer your question, no there was not conflict between Namibia and South Africa between 1939 and 1945.

Also, I will just take a moment to talk about British/South African cooperation during WW2. The situation was somewhat more complex than "(the UK) was pretty in charge of South Africa".

After the passage of the Statute of Westminster in 1931, the Dominions of the British empire (including the Union of South Africa) were granted legislative independence, although with the assumption that Commonwealth nations would come to Britains aid in the event of war.

However, in South Africa the decision to go to war in 1939 was not automatic. The prime minister, J.B.M. Hertzog, was in favor or neutrality, and his National Party faction was very skeptical of ties with Britain, partially due to lingering enmity because of the Boer Wars. Hertzog was opposed by WWI general and deputy-PM Jan Smuts and his partisans from the pro-british South Africa Party. (references to NP and SAP are somewhat anachronistic at this time, as both had united to form a United Party in 1934)

This struggle of wills was resolved at a party caucus, when Herzog lost control of the United Party to Smuts, and South Africa declared war on Germany on September 4th, 1939. Smuts would go on to serve as Prime Minister of the Union of South Africa until 1948, and South African forces would serve in campaigns in East Africa, North Africa, and volunteers served in the RAF.

But of course, support for Britain against Germany was not unanimous in South Africa. During the war, there were dissenters who advocated a pro-Nazi position, known as the Ossiewabrandwag (ox-wagon sentinels).

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Some recommended reading.

The First World War in Africa is a good, recent treatment of the war on the continent. He does address the Southwest Africa campaign, but also delves into the very important topic of Afrikaner nationalism.

Oxwagon Sentinel is another meditation on the subject of Afrikaner nationalism, this time exploring its more radical form during WWII.

South Africa History Online is a pretty good introductory resource to South African history. While there, I particularly recommend reading about Jan Smuts, who was a very important figure in South Africa in both World Wars.