Change Shape excludes the Spellcasting class feature
The last sentence of the Adult Bronze Dragon's Change Shape reads (emphasis mine):
Its statistics and capabilities are otherwise replaced by those of the new form, except any class features or legendary actions of that form.
Does Spellcasting count as a class feature? The 2018 D&D Basic Rules PDF says the following about monster special traits (p. 113; emphasis mine):
Spellcasting
A monster with the Spellcasting class feature has a spellcaster level and spell slots [...]
The monster has a list of spells known or prepared from a particular class.
Therefore, a dragon using Change Shape to become an Evoker does not have access to the Evoker's Spellcasting.
Rules sources disagree on "class feature" vs "special trait"
The latest digital sources agree that Spellcasting is a "class feature". The physical Monster Manual is a weird outlier, though. It went from "class feature" in the first printing, to "special trait" in the fifth printing*.
The November 2018 Monster Manual Errata PDF does not mention any changes, so the most likely scenario is that an error was introduced by the time of the fifth printing, due to copying from older digital sources (like the SRD).
Additionally, all sources include the line
The monster has a list of spells known or prepared from a particular class.
...which indicates that Spellcasting is a class feature. Lastly, Jeremy Crawford states that the Spellcasting trait in monsters is a class feature. His tweets are no longer considered official rulings, but they can still provide insight into the intent behind certain rules.
What follows is a breakdown of which sources say "class feature" and which say "special trait":
Class Feature
A monster with the Spellcasting class feature has a spellcaster level and spell slots [...]
Special Trait
A monster with the Spellcasting special trait has a spellcaster level and spell slots [...]
(Credit to V2Blast for pointing out these differences in the comments.)
* (The fifth Monster Manual printing is denoted by "9 8 7 6" below "First Printing: September 2014" on the back side of the first page. The first printing would be "9 8 7 6 5 4 3 2", the second printing would be "9 8 7 6 5 4 3", etc.)
If Change Shape didn't exclude Spellcasting, you would form with the necessary components
The Monster Manual has an entire section titled "Statistics". In it, the last subheader reads (MM, p. 11):
Equipment
[...]
If a spellcasting monster needs material components to cast its spells, assume that it has the material components it needs to cast the spells in its stat block.
Material components count as equipment, equipment counts as statistics, and Change Shape replaces the dragon's statistics, so the dragon would also form with any necessary material components.
(Credit to Rubiksmoose's answer to "What all is included in a creature's game statistics?".)