Summary (Nov 2021): It is possible for your child to apply later in Scenarios 1 and 2. For Scenario 3, it is also possible but normally your child has to submit their application together with or after your wife's application (i.e. not before) due to safeguarding provisions in the Immigration rules. Exceptions apply.
The main documents to consult on dependents of Skilled Workers are Paragraphs SW 26-36 of the Immigration Rules Appendix Skilled Worker (Immigration Rules hereafter) and Immigration Staff Guidance - Dependent family members in work routes (Guidance hereafter).
I am not a lawyer - the answer is based on a layman's interpretation of the rules. If in doubt, seek legal advice.
Both skilled worker and dependent partner outside UK & applied for/granted visa
Page 8 of the Guidance mentions:
Dependent partners and dependent children are not required to apply at the same time as the lead applicant, they can also apply to join the lead applicant at a later date.
...and thus it is possible for the dependent child (3 months old and thus clearly dependent age-wise) to apply later in this most straightforward scenario.
Skilled worker in UK (with visa), dependent partner outside UK & applied for/granted (dependent) visa
There are extra rules in the Immigration Rules that intend to act in the best interest and safeguard the welfare of the dependent child. The one that is at risk of conflicting with this scenario is Paragraph SW 30.2, which reads:
SW 30.2 The applicant’s [(dependent child's)] parents must each be either applying at the same time as the applicant or have permission to be in the UK (other than as a Visitor) unless:
(a) the parent applying for or with entry clearance or permission to stay as a Skilled Worker is the sole surviving parent; or
(b) the parent applying for or with entry clearance or permission to stay as a Skilled Worker has sole responsibility for the child’s upbringing; or
(c) the parent who does not have permission as a Skilled Worker –
(i) is a British citizen or a person who has a right to enter or stay in the UK without restriction; and
(ii) is or will be ordinarily resident in the UK; or
(d) the decision maker is satisfied that there are serious and compelling reasons to grant the child entry clearance or permission to stay with the parent who is applying for or has entry clearance or permission as a Skilled Worker.
A similar text is also provided in Page 20 of the Guidance. The intent of this paragraph is explained to immigration case workers in the next section of the guidance:
While you should normally be satisfied that a child is either joining both parents, or the work or study migrant is the sole surviving parent, or has and has had sole responsibility...
In this scenario, as both the skilled worker and the dependent partner has obtained permission to be in the UK other than a visitor, this particular rule is satisfied. Together with the text of the Guidance in the previous scenario, it is safe to conclude that it is possible for the dependent child to apply later.
Skilled worker in UK (with visa), dependent partner outside UK & not applied for visa
There are three cases that can arise from this scenario:
- The family proceeds to apply for (and assuming everything else is in order, is granted) a visa for dependent partner only, which leaves the family in Scenario 2.
- The family proceeds to apply for a visa for both the dependent partner and dependent child, which is effectively Scenario 1.
- The family proceeds to apply for a visa for the dependent child only.
As explained in the previous sections, the first two cases are fine. The third case is more problematic though, as normally it is in contravention with Paragraph SW 30.2 of the Immigration Rules - only one parent has permission to be in the UK.
Having that said, there are a few exceptions as listed in Paragraph SW 30.2(a-c). In addition, the immigration case worker can accept "serious and compelling reasons" under Paragraph SW 30.2(d). Page 20 of the Guidance include the following examples:
The following list is not exhaustive, but are examples of scenarios where it may be
judged that there are serious and compelling reasons to grant the child entry
clearance or permission to stay with the parent with permission:
- due to employment or study commitments, one parent is delaying their entry to
- one parent is travelling to a territory for reasonable purposes which makes it
impossible to care for the child e.g. the father/mother is travelling to a
country where it would not be safe for the child to join them
- one parent is travelling to a territory for reasonable purposes on a short-term
basis such that it would be unreasonable to expect care/schooling
arrangements to be made for the short length of the trip
- one parent is receiving medical treatment abroad and plans to join the family later