Is someone else's surgery a legally permitted reason for international travel from the UK?

This question would probably be a better fit for law.stackexchange.com. With that said, I’ll try to answer it from the legal perspective.

Firstly, guidance is not law. Guidance is instead usually (a) a summary of what the author of the guidance considered to be the law at the time of writing, and/or (b) a collection of recommendations which are not legally binding (usually phrased in terms such as you "should" do something, as opposed to you "must" do something). So, while the guidance is a useful starting point particularly for people without legal experience, it can (and does) contain things which are not completely accurate from a legal perspective. With that said, guidance can sometimes be useful in a court setting and I will explain below how this could be relevant for the coronavirus rules.

The legal position differs depending on whether you are in England, Wales, Scotland or Northern Ireland. As you didn’t specify which I will answer on the basis of England, but you can find answers for the other regions by looking at the equivalent regulations.

The rules for England are governed by the The Health Protection (Coronavirus, Restrictions) (All Tiers) (England) Regulations 2020 (as amended). The relevant part for our purposes is Schedule 3A (Tier 4 restrictions.

The starting point is paragraph 1(1):

No person who lives in the Tier 4 area may leave or be outside of the
place where they are living without reasonable excuse.

This is expanded upon in paragraph 1(2):

For the purposes of sub-paragraph (1) – the circumstances in which a
person has a reasonable excuse include where one of the exceptions set
out in paragraph 2 applies;

Here we have a very common source of misunderstanding. While paragraph 2 contains a list of exceptions, this is not an exhaustive list. This is clear from the wording of paragraph 1(2) above – circumstances of reasonable excuse include the exceptions but that does not mean there cannot be other reasonable excuses. The purpose of paragraph 2 is to provide a list of scenarios in which you definitely have a reasonable excuse (i.e. you cannot be prosecuted if your facts fall within one of those scenarios).

That leaves open the possibility of other reasonable excuses. Here, you take more of a risk since you do not have the protection of being covered by any of the exceptions. Instead, you would be reliant on how a court would judge the reasonableness of your actions. The word "reasonable" in a court usually refers to what is know as the objective test. This means that it isn’t based on whether you thought your actions were reasonable, but rather whether a so-called "reasonable person" would have thought so. This is where the guidance can be legally persuasive. If the guidance suggests that you should or should not do something then, even though it isn’t legally binding, it might persuade a court that it was reasonable or not reasonable to do that thing since a "reasonable person" would probably think it was reasonable to follow the guidance.

An approach which may increase the chance that a court would side with you is if your scenario is similar in spirit to the scenarios listed in the exceptions. Your scenario seems serious enough that it potentially could be a reasonable excuse. But as said above, you will always take some risk when doing something which is not explicitly listed as an exception.

Turning back to the exceptions, the exception in Schedule 3A, paragraph 2(7)(e) is probably closest to your case:

[Exception 4 is that it is reasonably necessary for P to leave or be
outside P’s home] to visit a person (“V”) receiving treatment in a
hospital or staying in a hospice or care home, or to accompany V to a
medical appointment and P is a member of V’s household, a close family
member of V, or a friend of V.

Note the words "reasonably necessary". If there is an alternative to you leaving your home, then it is arguably not reasonably necessary for you to do so.

The exception in Schedule 3A, paragraph 2(9) could also be a close match, depending on your mother’s prognosis:

Exception 6 is that it is reasonably necessary for P to leave or be
outside P’s home to visit a person (“D”) whom P reasonably believes is
dying, and P is a member of D’s household, a close family member of D,
or a friend of D.

Is visiting your grandmother right now even a good idea?

The UK is currently a major Covid hotspot, with a virulent new strain emerging there. Visiting a critically ill, elderly person right before a major surgery seems like a very bad idea to me. She’s already a member of the most at-risk group in society of dying or suffering serious complications from Covid due to her age; her illness and surgery just makes it even more risky for her.

I think that visiting her in person is a very bad idea, as a result. What if you were infected by the UK strain of Covid shortly before leaving the country, and then you gave it to her, and then she got sick and died? You’d have to live with the guilt of that for the rest of your life.

Would it be possible for you to see her over a session of Zoom or something instead? It’s not as good as meeting her in person, but it’s better than risking her health in such a critical moment for her.

Your need to see a person before a surgery, or to bring some people to see that person, is not a medical emergency. Her need for the surgery, may be, but it is not her who wants to travel. I doubt you will get an exception.

Where you might have a chance is, it says on your link that you can travel for the same reasons you can leave home, and you can leave home

to visit someone who is dying or to visit someone receiving treatment in a hospital, hospice or care home, or to accompany a family member or friend to a medical appointment

Now, it also says you can leave home to exercise, and I’m sure you couldn’t get permission to fly to another country to exercise, so this isn’t a guarantee of success, especially since this "major surgery" is one that can be postponed a few times already.

I can’t see a way for you to apply for permission to travel; it looks like you just do it and perhaps argue with someone at the airport about it? This looks too risky to me.

My personal definition of emergency travel (under which I did travel internationally in November) is: does any cost, quarantine or other issue mean I won’t go, or will I go no matter what? I will go no matter what, then I go. In my case this involved two separate two week quarantines, one of which cost me over $5000, and Covid testing. Didn’t matter, I had to go. If your need is not that urgent, then you know you should not go.