"Are street preachers worthy of being sectioned?"
No - so far as I'm aware, no police officer has tried to use this power with regard to a street preacher, it doesn't seem relevant.
To be involuntarily detained, compulsorily admitted, removed or 'sectioned' under s136 Mental Health Act 1983, "Removal etc of mentally disordered persons without a warrant", the person must appear "to the constable to be suffering from mental disorder and to be in immediate need of care or control" and the constable must think it's necessary "in the interests of that person or for the protection of other persons".
"Are the street preachers ... given undue privilege in their legal treatment due to the fact that they are propagating a religious cause?"
There have been numerous arrests, prosecutions and convictions (and successful appeals) of street preachers. The question at law in a particular case is whether, given all the circumstances (including the preacher's behaviour and the public's behaviour), the interference with the preacher's (Human Rights Act) rights is necessary, proportionate and in pursuit of a legitimate aim.
Every public authority (police, prosecution, courts, local authority etc) is required to give effect to the scheduled rights in the Human Rights Act 1998 unless statute makes that impossible. In this context, usually the two rights engaged are Articles 9 and 10.
Article 9 Freedom of thought, conscience and religion
Everyone has the right to freedom of thought, conscience and
religion; this right includes freedom to change his religion or belief
and freedom, either alone or in community with others and in public or
private, to manifest his religion or belief, in worship, teaching,
practice and observance.
Freedom to manifest one’s religion or beliefs shall be subject only
to such limitations as are prescribed by law and are necessary in a
democratic society in the interests of public safety, for the
protection of public order, health or morals, or for the protection of
the rights and freedoms of others.
Article 10 Freedom of expression
Everyone has the right to freedom of expression. This right shall
include freedom to hold opinions and to receive and impart information
and ideas without interference by public authority and regardless of
frontiers. This Article shall not prevent States from requiring the
licensing of broadcasting, television or cinema enterprises.
The exercise of these freedoms, since it carries with it duties and
responsibilities, may be subject to such formalities, conditions,
restrictions or penalties as are prescribed by law and are necessary
in a democratic society, in the interests of national security,
territorial integrity or public safety, for the prevention of disorder
or crime, for the protection of health or morals, for the protection
of the reputation or rights of others, for preventing the disclosure
of information received in confidence, or for maintaining the
authority and impartiality of the judiciary.
So speech and religion have some statutory protection but, like most of the Convention rights, they are 'qualified' rights. Their second paragraphs say when the state can lawfully interfere with the rights. For an interference to be lawful, it must be prescribed by law and necessary in a democratic society; proportionate (i.e. no more than necessary); in pursuit of a 'legitimate aim' in the second paragraph; and rationally connected to the legitimate aim.
In this context the legitimate aims are usually "the protection of public order" or "the prevention of disorder or crime" and the "prescribed by law" part are the Public Order Act and other statutory powers of the police. Interferences, infringements or infringing measures are things like dispersal notices, arrests, prosecutions, convictions and sentences (community orders, fines or imprisonments).
The escalation commonly goes from the police asking the preacher to choose their words more carefully or leave, then order to leave, then arrest.
The media has reported on a number of arrests, prosecutions, convictions and appeals. Try searching the web for "UK street preacher arrest". Be aware that some results are for websites with undeclared involvements with or links to defending the accused preachers.
I've put some names and links below and here is a freely available judgment that contains discussion of all the above, describing that process of weighing up that every part of the state must do when it infringes on the Convention rights: Overd & Ors v The Chief Constable of Avon And Somerset Constabulary  EWHC 3100 (QB) (19 November 2021). The judgment also cites other judgments to which it links, which may be of interest.
As backgound the four men, Overd, Stockwell, Karns and Clarke, were arrested while preaching at Broadmead shopping centre in Bristol in July 2016. The police had received complaints from the public about the content of the preaching. On arrival, the police officers perceived imminent public disorder.
The four were detained, then charged under s5 Public Order Act; the case against Karns was dropped before court and the other three were tried; Clark submitted there was no case to answer and he was successful; Overd and Stockwell were convicted but their convictions were overturned on appeal.
Subsequently, in December 2020 the four sued the police, alleging breaches of Articles 9, 10 and/or 11 of the European Convention on Human Rights, wrongful arrest, assault/trespass to the person/battery, false imprisonment, malicious prosecution and misfeasance in public office. Their claim was dismissed on all counts, they appealed and this judgment is in that appeal which was also dismissed.
Other examples (but not judgments, only media reports):
In Leeds in June 2021, David McConnell was preaching about "adulterers, drunkards, homosexuals," when he was challenged by a transwoman. McConnell repeatedly referred to her as a "man", a "man in woman's clothing" and a "gentleman". In September 2022 he was convicted of intent to cause harassment, alarm or distress, sentenced to a 12-month community order with 80 hours unpaid work and ordered to pay fines totalling £715. A spokesperson for the Crown Prosecution Service (CPS) said McConnell's words "crossed the line between a legitimate expression of his religious views, to become a distressing and threatening personal attack" (BBC).
In March 2023 his conviction was overturned. The court found there was no evidence McConnell intended "harassment, alarm and distress" (BBC).
In November 2020 John Dunn was preaching on Swindon High Street when two women, holding each other's hands, walked past him. He said, "I hope you are sisters," to which they replied that they were in a same-sex mariage. Dunn responded, "it says in the Bible that homosexuals will not inherit the kingdom of God." The women made a complaint to the police.
Dunn was subsequently charged under s5 Public Order Act. The CPS eventually dropped the case but not before causing some controversy when it was made public that a prosecution document sent to the defence said some parts of the Bible are "abusive" and "simply no longer appropriate in modern society and which would be deemed offensive if stated in public." The CPS later said this was not its official position.
In April 2021, John Sherwood was preaching outside Uxbridge Station in London when police officers approached him and said they had been told he had made homophobic comments. They arrested Sherwood under s5 Public Order Act for allegedly causing alarm or distress. (Daily Mail)
In April 2022, Sherwood was tried and acquitted. Apparently his defence centred on Article 10 (freedom of expression) - I cannot find a judgment.
In March 2018 a Canadian preacher was arrested outside Barking Station in London. A female passerby had complained to the police that he had made homophobic comments. After being detained for 20 hours, he was released without charge.
Some preachers are privileged in terms of their legal resources - some for example have been aided by the Christian Legal Centre (of Christan Concern), which has also involved itself in numerous cases ranging from 'gay cakes' to embryonic stem cell research. This Radio 4 documentary by Joshua Rozenberg, A Tale of Belief and the Courts, may be of interest.