In most immigration applications, a valid proof of ID is a mandatory requirement. Without an acceptable proof of ID, the Home Office may not even consider the application as valid. Exceptions to this requirement are set out in paragraph 34(5)(c) of the Immigration Rules.
Although the mandatory requirement may appear to be quite flexible, it may be problematic for an applicant to overcome this hurdle if e.g.
a. The applicant has an expired Travel Document from another country and s/he is no longer resident of that country and needs to renew their UK visa; or
b. The applicant has never held a passport and has no other photograph ID; or
c. The applicant cannot renew their passport in-country (i.e. the UK) and they have no other form of photographic ID.
Scenario A means that that some thought is needed to manage the timing when making the relevant applications. An applicant holding a non-UK travel document can apply for a UK travel document but only when they have 6 months or more limited leave on their visa. There are 4 types of Travel Documents - Convention Document, One-way document, Certificate of Travel and Stateless person's document. It is important to bear in mind that some countries may not recognise these types of documents and therefore it would not be possible for them to enter that country until they acquire a nationality and then a passport (which could take years to obtain). If the person's work requires travel, this will certainly be a factor to take into consideration.
Scenario B and C would prove to be a bit more tricky and will require some evidence gathering before the Home Office makes a decision whether the submitted application is valid. Immigration Rules paragraph 34(5)(c)(vii) allows the Home Office to exercise discretion where an applicant cannot submit an acceptable proof of ID provided that there is "… a good reason beyond their control why they cannot provide proof of their identity" . This may involve going through the motions of trying to obtain an acceptable proof of ID e.g. a passport and even be rejected by the High commission or embassy. This is so that physical evidence can be provided to the Home Office to show that there is a good reason they cannot provide proof.