A secular state does not mean that elected representatives are not allowed to publicly follow a religion or use their religious convictions to justify decisions they make. What it actually means is:
- Separation of religious institutions and the state. Religious leaders do not have any legislative, executive or judicial powers just based on their religious position. (Religious leaders usually do have the freedom of speech to take positions on political topics, but the politicians are free to ignore them)
- No state religion. Representatives and political officials are not obligated to follow a specific religion or any religion at all. For example, the US house of representatives includes 26 Jews, 3 Muslims, 3 Hindus, a Buddhist and 13 people who do not officially state their religious beliefs. According to the website of the UK parliament, the 2017 House of Commons included at least 15 Muslims and one Sikh. Political officials can choose if they want to swear their oath on the Bible, a different religious document, a secular document or no document at all.
- No religious document with judicial relevance. For example, the constitutions of many Islamic countries directly refer to the Quran as a source of constitutional and/or criminal law. A law can be declared unconstitutional if a council of theologians determines that it violates the Islamic Sharia law as written in the Quran.
Is the US a secular state? According to the First Amendment to the US constitution, it is:
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof;
That means that there is no state religion, but it also means that the US President is free to publicly exercise his personal religious beliefs by posing with a bible in front of a church. And US citizens are free to think of that what they want.
The official state motto is still "In God We Trust" which might imply that the state religion is monotheistic. But there is nothing in the US political processes which gives this phrase any powers.
Is the UK a secular state? Well, yes and no. While laws don't need to follow religious rules and while most elected politicians do not need to follow a specific religion, there are still 26 seats in the House of Lords which are reserved for Bishops from the Church of England. Further, the monarch of the UK (either the most or the least important politician in the country, depending on your point of view), is also the Supreme Governor of the Church of England and must be a Reformed Protestant.
Regarding the oath swearing ceremony depicted in the video in the question: The Oaths Act of 1978, which is still the current law for the oaths of
Members of Parliament, mandates that Christians and Jews have to swear (or affirm) "by Almighty God". According to this law, Christians swear on the Bible, while Jews swear on the old testament. So yes, the book depicted in the video is very likely a Christian Bible. But the law further says that:
"In the case of a person who is neither a Christian nor a Jew, the oath shall be administered in any lawful manner."
which means that MPs who follow different religions (or no religion at all) are free to deliver their oath in a form which fits their personal beliefs. For example, at the 2:00 mark, the video shows how MP Sajid Javid (who describes himself as not following any religion) delivers a purely secular oath:
"I do solemnly, sincerely and truly declare and affirm, that I will be faithful and bear true allegiance to Her Majesty the Queen Elizabeth, her heirs and successors, according to law."