Doom's beta, to which players who pre-order Wolfenstein: The New Order will be invited, is only coming to PC, PS4, and Xbox One which suggests that the game itself will also be available exclusively on those platforms.
In an updated FAQ on The New Order's website, Bethesda says that PS3 and Xbox 360 owners will be invited to the PS4 and Xbox One beta respectively:
"If you purchase an Xbox 360 or Xbox One version of Wolfenstein: The New Order, you will receive an invitation to the Xbox One Doom beta. PlayStation 3 or PlayStation 4 purchasers will receive an invite to the PS4 Doom beta. All PC purchases will receive an invite to the PC Doom beta. The Doom Beta will only be available on Xbox One, PlayStation 4, and PC."
Once you have the Doom beta code, you can access it through The New Order's website and Bethesda is working with Sony, Steam, and Microsoft to ensure that digital purcahsers receive beta keys.
An Xbox Live Gold subscription is needed for the Xbox One beta but a PlayStation Plus account is not required for the PS4 beta.
Bethesda has yet to reveal when Doom's beta will be active but says that it will take place some time after The New Order's launch. In addition, Bethesda states that the only way to currently access the beta is by pre-ordering The New Order which is set for release on May 20th in North America and on May 23rd in Europe on PC, PS3, PS4, Xbox 360, and Xbox One.