The answer to this lies in 8900.1, Volume 5, Chapter 2, Section 18 - Conduct an Airline Transport Pilot Certification, Including Additional Category/Class Ratings.
In most cases, yes the type rating "upgrades" or "elevates" along with your license since the type rating is always evaluated at ATP standards, even if you only have a private or commercial certificate.
Exceptions are type ratings for single-engine airplanes and type ratings that have a "VFR ONLY" restriction.
8900.1 excerpt:
5-710 PILOT CERTIFICATE LEVEL AND CATEGORY AND CLASS RATING. The following category and class rating(s) for which the applicant has
qualified on the original or subsequent ATP practical tests are
entered on the ATP certificate.
A. Addition of Lighter-than-Air and Glider Class Ratings. A lighter-than-air or glider category rating may be added to an existing
ATP certificate; however, the rating may be added only at the
recreational, private, or commercial pilot certificate level, as
appropriate to the practical test completed. There is no provision for
the original issuance of an ATP certificate with a lighter-than-air or
glider category rating.
B. Type Ratings. Type rating tests are conducted to the ATP standard for all grades of pilot certificate. Therefore, all type
ratings in that category and class of aircraft for which the practical
test is conducted in are upgraded after successful completion of the
practical test. For example, a person who holds B-737, DC-3, and SK-62
type ratings on his or her commercial pilot certificate, and later
satisfactorily completes an ATP practical test in a CE-750, would have
his/her B-737 and DC-3 type ratings elevated up to the ATP
certificate. The SK-62 would not elevate up because the practical test
was in a multiengine airplane. However, when the person satisfactorily
completes an ATP practical test in a helicopter, the SK-62 would then
elevate up to the ATP certificate. Therefore, all of the type ratings
held on the superseded certificate carry forward at the new
certificate level within category and class.
NOTE: A type rating for a single-engine airplane may not be upgraded
to the ATP level.
1) Except for type ratings and, under some circumstances, the
instrument rating, other ratings indicated on the superseded pilot
certificate are carried forward at the commercial, private, or
recreational pilot certificate level, as indicated on the superseded
certificate.
2) Instrument rating privileges are shown on the ATP certificate
only if the ATP practical test was conducted under visual flight rules
(VFR) only, or to retain instrument privileges that were held on the
superseded certificate for a category of aircraft other than the one
used for the ATP practical test.
C. Type Rating Limited to VFR. A type rating bearing the limitation “VFR ONLY” may be added to an existing certificate other
than an ATP certificate under the provisions of § 61.63(e). This
limitation may be added to an aircraft type rating not capable of
instrument maneuvers or procedures required on a practical test. The
practical test must be administered under the appropriate ATP PTS for
an aircraft type rating limited to “VFR ONLY”. A type rating bearing
the limitation “VFR ONLY” may be added to an existing ATP certificate
under the provisions of § 61.157(g). The practical test must be
administered under the appropriate ATP PTS for an aircraft type rating
limited to “VFR ONLY.”
NOTE: There is no provision for taking an initial practical test for
an ATP certificate in an aircraft that would require a VFR limitation.
I don't have to retake the instrument checkride when I get my commercial license. But you do have to demonstrate instrument tasks. If you don't, your licence will be limited to VFR only. $\endgroup$