In the US, the FAA's requirements for obtaining authorization to fly former ,military aircraft is included in 8900.1 Volume 5, Chapter 9, Section 2.
It is quite lengthy and covers more than just former military aircraft. As others have pointed out, most of these aircraft do not have a Type Certificate. They normally have a Special Airworthiness Certificate in the category Experimental Aircraft Exhibition. Requirements to fly the aircraft are spelled out in the aircraft's operating limitations which are issued with the Special Airworthiness Certificate.
From 8900.1:
A. Background. The FAA requires pilot authorizations to operate
some aircraft in the experimental category, including some former
military, all turbojet‑powered (for this section, all turbojet‑powered
includes all turbofan‑powered), all rocket‑powered, and all large
(over 12,500 pounds maximum gross takeoff weight (MGTOW)) aircraft,
and any other aircraft requiring specific pilot skills. Examples of
such models are the Northrop F‑5, Bell P‑63, MiG‑15, Ju‑52, Mi‑24, and
the BD‑5J. The FAA also requires authorizations to act as PIC of large
or turbojet‑powered aircraft undergoing type certification.
1) The requirement for an FAA authorization to act as PIC of
certain experimental aircraft is contained in the FAA‑issued aircraft
operating limitations. Pilots are required to comply with the
FAA‑issued operating limitations by 14 CFR part 91, § 91.9(a).
2) Because these aircraft are not type certificated (TC), a type
rating is not available. In the absence of type ratings for these
aircraft, it is the FAA’s objective to ensure, for the pilots flying
these aircraft, a level of safety and proficiency similar to what is
available for an aircraft with a type rating.
The specific pilot requirements are also spelled out in the document:
A. Former Military Turbojet/Turboshaft‑Powered Aircraft. To be
eligible for an authorization to act as PIC of a former military
turbojet/turboshaft‑powered aircraft, an applicant must:
1) Possess at least a U.S. private pilot certificate with an
appropriate category and class rating for the configuration of the
aircraft;
2) Hold an instrument rating;
3) Possess at least a valid U.S. third‑class medical certificate or
equivalent (U.S. Military Flight Medical or U.S. driver’s license in
accordance with the FAA BasicMed process (pilots using BasicMed may
not exceed 250 knots indicated airspeed (KIAS)));
4) Have logged a minimum of 500 hours of pilot flight time in the
aircraft category and have completed the U.S. armed services
qualification checkout described in this section; or have logged a
minimum of 1,000 hours pilot flight time, including 500 hours as PIC
in the aircraft category, and have completed the training requirements
of this section; and
5) If the aircraft is capable of supersonic flight, have a minimum
of 250 hours of pilot flight time as PIC of a fixed‑wing
turbojet‑powered aircraft, in a Group V, VI, or VII aircraft (see
Figure 5‑173), or present proof of completion of a U.S. military
qualification in a supersonic turbojet‑powered aircraft.
B. Former Military Propeller‑Driven Airplane. To be eligible to
serve as PIC of a former military propeller‑driven airplane that has a
MGTOW exceeding 12,500 pounds, or which has a horsepower rating of
more than 800 horsepower and a VNE that exceeds 250 knots, an
applicant must:
1) Possess at least a U.S. private pilot certificate with an
appropriate category and class rating;
2) Possess at least a valid U.S. third‑class medical certificate or
equivalent (U.S. Military Flight Medical or U.S. driver’s license in
accordance with the FAA BasicMed process (pilots using BasicMed may
not operate an aircraft with a MGTOW above 6,000 pounds));
3) Have logged a minimum of 500 hours of pilot flight time; and
4) Have completed the training requirements of this section.
That last item can be a challenge as it requires finding someone with the proper authorization to provide the training.
More information is available at the FAA Vintage & Experimental Aircraft Program web page.