This is likely partly a matter of style rather than strict grammar.
In terms of the use of neither-nor, The Chicago Manual of Style (17th ed.), 5.234, says the following:
The correlative conjunctions neither and nor negate alternatives simultaneously. Traditionally, only pairs are framed by neither–nor, but writers and speakers sometimes use a neither–nor–nor construction, as in the last example below.
The dog and the cat are not friendly.
Neither pet is friendly.
Neither the dog nor the cat is friendly.
The radiator does not leak, and the water pump also does not leak.
Neither the radiator nor the water pump leaks.
Neither John nor Sally nor Brenda can attend the meeting.
In that last example, some writers include only the last nor. But again, a simple neither–nor construction isn’t recommended with three or more elements, the sequence neither–nor–nor being preferable.
Perhaps the most famous of multiple list items using a neither-nor construction is the one mistakenly thought to be the official motto of the US Post Office:
Neither snow nor rain nor heat nor gloom of night stays these couriers from the swift completion of their appointed rounds.
Note that when you use neither-nor (or either-or) the verb used follows the singularity or plurality of the individual list items.
Based on this guidance, and with the three list items, the sentence in the question could be phrased in the following ways:
Neither hair dryer nor shower gel nor shampoo was in the room.
There was neither hair dryer nor shower gel nor shampoo in the room.
As a matter of style, you could also use commas to separate the list items.
Other variations of the sentence that don't use neither-nor include the following:
There wasn't a hair dryer, shower gel, or shampoo in the room.
There was no hair dryer, shower gel, or shampoo in the room.
There wasn't a hair dryer or any shower gel or shampoo in the room.
You could also reverse the sentence, which might sound more natural than any of the other forms:
The room contained no hair dryer, shower gel, or shampoo.
The room didn't contain a hair dryer, shower gel, or shampoo.
There is no single correct version. It's a matter of style and what you think sounds best.