The question refers to offences under sections 4A or 5 of the Public Order Act 1986, which respectively define the main elements of the offences by:
A person is guilty of an offence if, with intent to cause a person harassment, alarm or distress, he—
(a) uses threatening, abusive or insulting words or behaviour, or disorderly behaviour, or
(b) displays any writing, sign or other visible representation which is threatening, abusive or insulting,
thereby causing that or another person harassment, alarm or distress.
and
A person is guilty of an offence if he—
(a) uses threatening or abusive words or behaviour, or disorderly behaviour, or
(b) displays any writing, sign or other visible representation which is threatening or abusive,
within the hearing or sight of a person likely to be caused harassment, alarm or distress thereby.
For a person's conduct to meet the definition of either offence, it is not enough that it causes, or might cause, "harassment, alarm or distress". Under 4A it must be done with that intent, and there are additional intent requirements in section 6. And either way there must be the elements of threat, abuse or insult, as specified. There are other conditions and defences outlined in the statute, in particular for when the accused person can show "that his conduct was reasonable". Reasonable conduct must be interpreted consistently with the person's Convention rights under Articles 9, 10 and 11 (freedom of religion, free speech, freedom of assembly), because of the Human Rights Act 1998 - but prior to its enactment, equivalent common-law principles applied in any case.
So the brief answer to "can one's harassment alarm or distress (or that of many) based simply on the content of views being expressed render their expression a public order offence?" is no, because the other elements of the offence have to be made out, including the mens rea. And even if all that is present, the reasonableness defence can cover many examples of the expression of political or religious ideas.
The case law includes plenty of examples of what has and hasn't met the tests of "threatening", "abusive", or "insulting"; the intent; the extent of any actual distress; and the countervailing arguments relating to religious and political speech. Here are some. These are all appeal cases because magistrates' courts judgments don't usually get published, and aren't precedential.
"abusive": Swear words and racist terms are typically abusive, as in Taylor v DPP [2006] EWHC 1202 (Admin) (note: several such terms appear in the linked judgment).
"insulting": Hammond v DPP [2004] EWHC 69 (Admin) concerned the claimant's conviction under section 5, which at the time also included the "insulting" possibility. He had been displaying a sign in Bournemouth town centre with the words "Stop Homosexuality", "Stop Immorality" and similar language. This was deemed to be insulting, but not threatening or abusive, and it did cause distress for several people. The defence of reasonableness was rejected in this instance because his behaviour in the event "went beyond legitimate protest, was provoking violence and disorder and interfered with the rights of others".
"thereby causing ... distress": In R v DPP [2006] EWHC 1375 (Admin), the conviction of a boy under section 4A for calling police officers "wankers" was quashed. The word was abusive and insulting, but there was no evidence that he intended to cause distress (as opposed to annoyance), and the officers were offended but not particularly emotionally moved.
As to the Convention rights,
it must be recognised that legitimate protest can be offensive at least to some – and on occasions must be, if it is to have impact. Moreover, the right to freedom of expression would be unacceptably devalued if it did no more than protect those holding popular, mainstream views; it must plainly extend beyond that so that minority views can be freely expressed, even if distasteful.
(Abdul v DPP [2011] EWHC 247 (Admin) at 49(ii), but read the rest of the paragraph). Further, the Human Rights Act directs that courts give "particular regard to the importance of the Convention right to freedom of expression" (s.12(4)) and religious freedom (s.13(1)). Interference with these rights can be justifiable on the grounds of public order, and the assessment depends on all the circumstances. Thus in Percy v DPP [2001] EWHC 1125 (Admin), the defendant's conviction for trampling an American flag at a protest was quashed because the original judge had not considered Article 10 properly. But in Norwood v DPP [2003] EWHC 1564 (Admin) the display of an anti-Muslim poster "had not formed part of an open expression of opinion on a matter of public interest, but had become disproportionate and unreasonable", so the conviction stood.
