The English subjunctive is used sparingly by native speakers (some never use it) and is much less often used than in French, Spanish, or German.
The subjunctive has two forms:
The plain form of the verb, identical in form to a bare infinitive - some people prefer to reserve the term "subjunctive" for this form. It is found (a) in a few set phrases such as "so be it" and "long live the Queen", and (b) especially in American English (but sometimes in British English too) in subordinate clauses after certain verbs (primarily of ordering or suggestion), e.g. "he demanded that it be done".
Identical in form to the simple past, except for the verb "to be", for which "were" is used where "was" would normally be found. Some call this isolated form the "irrealis".
Lexico summarises the uses of the subjunctive here: https://www.lexico.com/grammar/when-to-use-the-subjunctive
The irrealis "were" is found after "if", "as though", and after "wish" in phrases such as "I wish that it were" (occasionally, the semi-archaic form "I would that it were" is seen or heard). It is also used in inversions ("were it a hot day" = "if it were a hot day").
It is difficult to think of any situation where "once" or "she assumed that" or the two combined would trigger the subjunctive. In the unlikely event that they did, it would be optional.
It is always better to err on the side of using the indicative (not the subjunctive). Otherwise you may fall into the trap of hypercorrection. Most people consider it acceptable to replace irrealis "were" with "was" in most cases, but not the other way round. (The linguist Geoff Pullum claims that even in the case of "if I were you", "if I was you" is "universally" accepted as an alternative - see
https://languagelog.ldc.upenn.edu/nll/?p=3038 - though this may be an overstatement.)
In your sentence
she assumed that once there was a new film everybody would be eager to go and see it.
The "once"-clause isn't part of the assumption as such. Unlike "if", the word "once" presupposes that a new film will be released. At any rate the supposition is strong enough that the subjunctive isn't an option, and "was" is the sole correct option here.
You asked why we use 'would' and yet can't use a subjunctive 'were' here.
We can say "If a new film is released, people will go to see it" and we can also say "If a new film were released, people would go to see it". But while we can say "Once a new film is released, people will go to see it", we can't say *"Once a new film were released, people would go to see it" (nor does the version with "was" work, except when discussing the past!). So "once a new film was released, people would go to see it" is the temporally backshifted version of "once a new film is released, people will go to see it". It is backshifted because we are discussing a belief that was held in the past, not backshifted as a form of modal remoteness as in "If it were...".
(You can say "If he gave me a job I would be happy to start immediately", but you can't say (as a complete sentence discussing a hypothetical present/future) *"When he gave me a job I would be happy to start immediately" - unless you are discussing the past, e.g. "I said that when he gave me a job I would be happy to start immediately". "When" and "once" don't allow for remote or counterfactual conditionality.)
The original sentence spoken must have been "once ... is..., ...will...". You can't say "Once/when X was/were..., ...would...". "Would" in this sentence is a backshifted "will" rather than a true conditional.
You can say "If X happened, I would do Y" as a reference to a hypothetical condition X, but you can't say "When/once X happened, I would do Y", unless you're discussing the past (e.g. "Whenever X happened, I would [=used to] do Y") or recounting a past statement (e.g. "I said that when X happened, I would do Y").