The short answer is that we don't know but we can hazard some guesses based on a comparison with Sudley House assuming that both were designed by the same architect, John Whiteside Casson (or "Daddy" Carson as he is called in a memoir about the original owner, Nicholas Robinson) with internal features common to both. Note that Sudley presently shows the art collection in the context of its later owner, wealthy Liverpool merchant and philanthropist George Holt, who made significant alterations and extensions to the original building.
What little information we have about the external appearance of Crown Street, arguably the first purpose-designed railway terminus, depends on images by Thomas Talbot Bury and Isaac Shaw. The results are instantiated in an OpenSimulator model which is presented as a work-in-progress.
Crown Street station c.1831 by Bury (left; Wikipedia) and Shaw (right; Yale Center for British Art)
The Dining Room at Sudley corresponds to the booking (or coaching) office at Crown Street
Interestingly, the relevant parts of both buildings are oriented east-west and we start in the west. For Crown Street this would have been the entrance courtyard looking into what I assume is the booking office.
Top: the model of the booking office; below: similar views of the Dining Room (NB position of fireplace obscured by door to left in image at top left)
The original part of the south wing at Sudley comprises two rooms currently called the Dining Room and the Drawing Room. Each has its own chimneystack shared, I would guess, with the room above on the first floor. The portico entrance to the station booking office would go through the middle of the wall dividing the Morning Room, a later addition at Sudley, from the Dining Room/booking office. The door at Sudley is currently towards the south side of the dividing wall but apparently both this door and the equivalent one on the far side of the Dining Room (leading to the Drawing Room/waiting rooms) are late additions. Normally people at Sudley would have used the doors to the hallway located to the north on either side of the fireplace. Any equivalent door at Crown Street would take you into a (private?) yard with sheds; there is no hallway. Indeed, the model excludes north-facing, ground-floor windows onto the yard for this largely aesthetic reason.
In both buildings the chimneystack and fireplace for this entrance room is located on the north side. To my mind this determines the layout with the desk for the booking clerks directly in front as passengers enter with two doors behind them, much as at Manchester Liverpool Road (which was built after Crown Street).
Sudley has an elegant staircase in the hallway. This isn't an option at Crown Street so my guess would be that there was a door located behind the desk and to the left/north that led upstairs to the offices. The door isn't strictly necessary but would serve as an additional security measure. The staircase at Manchester differs in being passenger-facing and is more elegant as a consequence. As an aside, the model originally had a Manchester-style staircase in the central room but this has now been removed.
The passenger experience
The door to the right/south of the desk leads to two waiting rooms, one beyond the other. The first corresponds to the Drawing Room at Sudley and the fireplace is on the far side of the dividing wall with the Dining Room, behind the desk area.
Initially railway travel was similar to present-day aviation; you booked in advance and checked-in before departure on the day of travel. On first class trains each passenger was booked to a numbered seat either at the Dale Street office or at the less central Crown Street. First class passengers had the option of taking a horse bus from Dale Street to the station. As these would arrive close to the time of departure it would likely have been a busy and confused scene with porters stowing luggage on the roofs of the carriages and passengers going straight to the train after check-in. Those who arrived by their own carriage, by cab or on foot could miss some of this bustle by checking-in early and sitting in the waiting room while the train was marshalled.
The first/westernmost window at Sudley is a door at the station and, once checked-in, passengers could go through that door to the platform if they so chose. A series of warning bells was used to signal departure and the passengers in the waiting rooms probably had the option of using an alternative door at the opposite (eastern) end to get to the platform. This end of the building does not exist at Sudley which has five window bays, three for the Dining Room, two for the Drawing Room. Images of the station by Bury and Shaw give no information on the number of bays or the presence of this second door to the platform although others, possibly with a degree of artistic licence, show a door and varying numbers of windows.
Given subsequent practice at Lime Street, it seems likely that passengers could purchase a newspaper or an orange from sellers permitted onto the station.
One interesting question is whether the terrace at Sudley inspired the platform at Crown Street. The latter probably looked not unlike the verandah at Sudley whose canopy, however, dates to much later. The early stations typically had low or non-existent platforms; the one at Manchester was not more than a few inches high. Shaw's picture of Crown Street, however, shows an elevation of the order of two feet (compared to the standard three feet nowadays) and explains the presence of a step from the yard to the platform as well as the steps up to the main door.
Whether Sudley originally had a terrace before it acquired the canopy is unknown. It is in any case a rather strange feature given that there is no door onto it from the house.
Top: Proposed model of Crown Street in 1830 (left) and 1831 (right)
Bottom: External view of south wing at Sudley, including non-original verandah.
As with his father before him, John Foster Jnr was a well-known Liverpool architect and contractor as well as being corporation surveyor. Several people have credited him with the station building and he is known to have designed the famous Moorish Arch in the nearby Chatsworth cutting. He was also responsible for the train shed at Crown Street. Plans for this were approved two months after the railway opened to passengers. Its dependence on the verandah canopy suggests that this might also be an addition by the same architect and, in the same vein, quite possibly the portico as well. By this time the directors would have appreciated the significant revenue being generated by passenger traffic on a line whose primary purpose was freight. The pillars supporting the canopy may, however, have made access to the carriages awkward and necessitated passenger use of the trackside doors.
There are significant imponderables such as the number of rooms on the ground floor. The model proposes three equally sized rooms on the basis of the number of chimneystacks and window bays but, of course, there is no bar to having a booking office and then a single large waiting room with multiple fireplaces.
One possibility is that the third room might have been for the use of second class passengers. At Manchester Liverpool Road the facilities for first and second class passengers were entirely separate. Crown Street, however, had less space available and passengers of the two classes may have been required to share waiting rooms. In practice they used different trains running at significant intervals so this could be managed. One interesting possibility that this raises, however, is that Crown Street was designed before the decision to operate two classes of train had been taken.
A fruitful proxy?
Whether one accepts that Casson's wing at Sudley was the prototype for Crown Street is perhaps secondary to the issues raised and the ways in which their resolution assists in refining the model. The model in turn informs our understanding of the way the station may have operated.
Railways were initially the province of the well-to-do. Adaptation of a merchant's "palace" into the first station may have been subconsciously reassuring to early passengers even if the experience was pared down and an ulterior motivation on the part of developers was convenient access to a proven design.
Demonstrating an unequivocal rather than circumstantial connection between Sudley and Crown Street will likely prove difficult. Nevertheless, there is now a new avenue to explore a question that has vexed many authors in the past. With the bicentenary of the railway in 2030 and the bicentenary of Crown Street likely some time before that (2028?), there may be an opportunity to develop an additional theme at Sudley to run in parallel with the Holt bequest.