In both English and German, the World Wars are, of course, numbered. There is a first one and there is a second one. The main difference between the two languages here, and the one that leads to what you perceive as different terms in English, is that English is totally fine with having a cardinal number follow an expression to denote a similar idea as an ordinal number preceeding it. Thus, from my point of view, the First World War and World War I (the Roman numeral I being used instead of the word One or the number 1) are equivalent, and the last ‘term’, WWI is merely an abbreviation of the second.
German does not really have this tendency. Instead, ordinal numbers are typically strictly placed before the expression they order. Thus, the direct correlate to First World War — Erster Weltkrieg — is in common usage while Weltkrieg 1 — the correlate to World War One — is unused and sounds weird.
Both languages have their conventions on how to typeset the number; as I mentioned before, English tends to use Roman numerals for the cardinal numbers following the expression. In German, there is no real difference between spelling out the ordinal number as Erster Weltkrieg or setting it as a numeral as in 1. Weltkrieg. Note that the numeral, being an ordinal number, carries an obligatory full stop and that the spelt-out word must decline according to case and presence/lack of the definite article as any other German ordinal number must.
Oftentimes, Germans are taught to spell out the numbers 1 to 12 and to use digits for the remaining ones and they follow that rule strictly. Thus, I would assume Erster Weltkrieg to be more common that 1. Weltkrieg unless there is a setting that greatly suggests enumeration in digits. This is confirmed by the following NGram of 2. Weltkrieg versus Zweiter Weltkrieg (searching for 1. Weltkrieg yielded no results):
Note that this analysis is already inherently skewed to the benefit of the digit, since Zweiter Weltkrieg is restricted to nominative without article.
