It was normal that a new emperor would be designated to become
consul ordinarius on the first 1 Jan. after
his accession, but I think it is probably never recorded exactly when and by what mechanism he received that designation.
The consular designation was so regular that one could suppose it was normally included among the titles and honors granted to the emperor on the day of
his accession. But maybe the senatorial decree
had to be confirmed by the vote of a popular assembly before the designation was considered official and could be recorded on coins.
We hear of consular assemblies meeting in March and again in October: see
Strack Trajan p. 18, note 40, referring to Mommsen's Römisches Staatsrecht and other sources. But maybe there was a way for the emperor to get around having to wait for the next regular assembly before calling himself
consul designatus.
Tacitus records that in 51 AD
Nero assumed the
toga of manhood, and that on this occasion
Claudius gladly acceded to the Senate's adulatory proposals that
Nero should become
consul when he reached the age of 20 (i.e. 7 years later, in 58 AD), and that in the meantime, as
consul designatus, he should have proconsular imperium outside
Rome and should be called Prince of the
Youth. This and other similar cases show that it was in the Senate's power to designate consulships, but we are never told whether or not confirmation from a popular assembly was necessary.
Anyway, I think you are certainly correct that the quadrantes of
Claudius with titles
PONT MAX(I)
TR POT IMP were the earliest of the reign, and because of their
rarity can only have been struck for a short time.
I think only two such quadrantes are known today: one in
BMC 173, pl. 35.12 with MAXI, and one in
Munich with
MAX. See
von Kaenel,
Claudius, p. 27.
Cohen 75 records a specimen with
MAX from a private
collection, which will be a third specimen whose whereabouts today are unknown, unless it's the same coin as the specimen now in
Munich, or as the BM coin (acquired in 1913) with the unusual reading MAXI overlooked.