As an example, imagine an ordinary Jonathan Reez residing in London. Would it be a civil or criminal offense for Jonathan to ask others to call them Sir Jonathan Reez, print business cards with that name, use it on Linkedin, etc? In other words, is the title "Sir" legally protected in the UK?
Let's look at the most recent directive on this topic, gazetted on 1 June 2016:
Only those British nationals, including dual nationals, awarded a British Knighthood or appointed to a British Order of Chivalry as a Dame, may use the title ‘Sir’ or ‘Dame’ in the United Kingdom.
Note that while the notice starts by talking about foreign titles, it actually establishes a general rule. The link is that the significance of British honours is partly due to their rarity, and historically there have been plenty of people who've tried to buy knightly or noble titles from abroad (e.g. from cash-strapped former European monarchs). The "long-standing convention", in place for at least a couple of centuries, has been to avoid official recognition of foreign titles given to British subjects/citizens, except by case-by-case royal permission.
A first observation is that it is not a criminal offence to call yourself Sir. If it was, then a notice of this kind would be redundant. This notice, which like all other rules relating to orders of knighthood is a matter of the royal prerogative, does not and cannot create a criminal offence. There may be cases where using the title is part of an offence, most likely committing a fraud, but Sir is not among those titles that are protected in law. As a contrasting example, falsely using a title of Doctor in a medical context is an offence, because there is a statute making it so. In the same way, there are certainly ways to get in trouble in the civil courts, but simply introducing yourself as Sir Jonathan does not injure anyone.
One reason why this is not an offence is that there are very few privileges pertaining to knighthood - it's basically about social recognition. There is not much reason to ban the practice of pretending to be a knight, especially since it is easy these days to check whether someone holds a genuine title. Remaining privileges might include the use of certain chapels for members of particular chivalric orders, and gaining a place in the official order of precedence at courtly functions - both of which are contexts where it is difficult to sustain the illusion.
The scope of the notice, then, is declarative of royal policy with respect to matters of precedence, the grant of arms, etc., and of government policy about recognition of titles, such as on a passport. Many people who adopt spurious titles are doing so not just because they want to call themselves Sir, but because they want some form of official blessing for it (hence the practice of buying a title from some random Internet microstate, which carries no more legitimacy than making the title up yourself, but somehow "feels" slightly more real).
According to the bottom of the How to change your name page of the Deed•Poll•Office website1:
There is no legal basis for changing your title if you haven’t acquired it legitimately — to Lord, or Sir, for example. We do not issue change of title deeds, because no-one is obliged to recognise your title, and a change of title deed is unlikely to make any difference.
It is possible to change your first forename to something like Lord or Sir, to give the impression that you hold that title, provided it is not for a fraudulent purpose. However you should think twice before doing this — you will find it difficult to get official bodies to accept the change of name. For more information, see our advice about presumed titles on passport applications.
So, you cannot just "call yourself" a Sir, but you could (against advice) change your name to Sir Reez instead of Jonathan Reez.
1 From their "who we are" page, "Deed Poll Office is a firm of paralegals specialising in change of name deeds (deed polls) and U.K. name change law.". I have no affiliation with them, or knowledge of them prior to seeing this question.
Telling lies is not illegal
Telling lies in order to obtain an advantage is fraud.
Whether styling yourself as “Sir” crosses that line depends on the circumstances.
There are protected titled in the UK
You can’t call yourself an architect, or a doctor, or a physiotherapist, or a lot of other things if you don’t hold the appropriate qualification.
However, calling yourself Sir Jonathan or a knight, is not one of these.