Responsibility for health insurance generally lies with the state where you work, and otherwise with your state of residence; nationality does not matter.
Anyone employed in Austria as an employee is subject to compulsory insurance in statutory health insurance, from the day employment begins even without the registration that the employer makes before work starts; self-employed people are subject to compulsory insurance with the SVS (social insurance institution for the self-employed), for example tradespeople who are members of the chamber.
Mere residence in Austria does not establish compulsory insurance: anyone who does not work, receives no pension or benefit from the Arbeitsmarktservice (Austrian public employment service) and is not co-insured can insure themselves with the ÖGK.
Anyone who is not subject to compulsory insurance in any statutory health insurance can insure themselves in Austria on application, as long as their place of residence is in Austria (§ 16 ASVG, Austrian General Social Insurance Act). The ÖGK names as a condition that you are not subject to compulsory insurance either in Austria or in an EU/EEA state.
Self-insurance follows seamlessly from a previously existing Austrian insurance if the application is made within six weeks of its end; otherwise it begins on the day after the application.
By law, self-insured persons are entitled to benefits only after a waiting period of three months, which the statutes can extend to six months; the ÖGK names six months, which do not apply if statutory health insurance in Austria, an EU/EEA state or a treaty state existed for 26 weeks in the previous twelve months or for six weeks immediately before.
Self-insurance provides benefits in kind but no sickness or maternity allowance, and ends, among other things, when the place of residence is moved abroad or when contributions are two months in arrears. Whether prior German insurance, statutory or private, affects the start or the waiting period is something you clarify directly with the ÖGK before the move.