Different types of legislation have different levels of approvals required.
Some things, like Section 3(1) registration of children as British citizens, are at the SSHD's discretion. They can just be guidance documents or similar and don't even really count as legislation. Home Office documentation essentially formulates the law/requirements for such topics.
Statutory Instruments (SI), which is essentially law-making delegated by Parliament to ministers (therefore it is also called delegated legislation), can take effect either automatically or can be subject to a negative or affirmative procedure. The procedure that is to be followed is spelt out in the Act of Parliament that gives the minister the power to make the SI/delegated legislation. The negative procedure means that if the minister signs off on an SI, it becomes law automatically unless either one of the two Houses of Parliament prays/votes against it. The affirmative procedure is that the both Houses of Parliament must approve the SI for it to come into force.
The Immigration Rules and its amendments, and the amount of fees payable for immigration and citizenship applications, are made by ministers under the negative procedure. So, unless either House of Parliament actively votes against it, it automatically becomes law. In actual fact, it is very rare for a prayer/vote against an SI in either House of Parliament. In case of the Commons, generally the government has a majority in the House and so can win a vote if there is one and the time of the House and which debates to hold is decided by the government, which holds a majority in the House. In case of the Lords, they are deeply conscious that they are unelected and that the law has been signed off by a minister that is responsible to the Commons. So they rarely vote against it.
The number of times that an SI has been prayed/voted against since WWII in either House is in single digits. So effectively the minister's signature on an SI subject to the negative procedure makes it law.
However, SIs are still subject to the courts. And
there has already been a court case about whether the Immigration Rules can be retrospectively changed for somebody who entered the UK under a certain set of rules. The courts ruled that once migrants had entered the UK under a specific set of rules, they had a legitimate expectation to continue under those set of rules till settlement. Note that this does not apply if the migrant changed their visa type/pathway to settlement (from skilled worker to spouse, for instance). But this judgement is the reason why changes in the Rules have so far only applied prospectively, not retrospectively.
However, this ruling, by definition, only binds SIs/delegated legislation made by ministers. An Act of Parliament (primary legislation) can override a judgement and make retrospective changes to the law. I understand that the Home Office plans to introduce primary legislation to implement the proposed changes. And that can change the rules even for people already in the UK.
It is worth remembering that in the UK, Parliament is supreme and can overrule the courts by changing the law, unlike other countries where the courts can override the legislature based on their rulings and judgements of a codified constitution.
So to quickly summarise, the changes to the Immigration Rules prospectively (for new migrants) can be brought in any time now, by statutory instrument and without a vote in Parliament. That will likely not affect the existing migrants. This excludes things like suitability criteria.
With regards to changes for migrants already within the UK, the battle for immigration control and changes to the Immigration Rules will move to Parliament likely in the next session of Parliament, after the local elections in May. If the Reform Party does well, that will increase the political pressure on the current UK government to make even more changes through an Act of Parliament, which can be retrospective.
With regards to the rush to the door to get British citizenship, keep in mind that even the rights of British citizens to live in or return to the UK can be curtailed. And they have been in the past. In the 1960s and 70s, the rush of CUKCs from the colonies to the UK as their countries gained independence led to a series of restrictive laws that prevented even people born with CUKC status outside the UK from entering the UK without a visa or permit. Remember that there is no restraint on the power of the UK Parliament to change the law, either prospectively or retrospectively. So do not assume that British citizenship will provide any level of security if the political climate in the UK changes further against immigration. We will always be reliant on the goodwill of the other citizens of the United Kingdom.
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.