This question has been inspired by the fact that Prince Charles has made comments about the UK government's new immigration policy of sending immigrants to Rwanda for processing. He called the policy "appalling".
As he is first in line to the throne, he may make such comments in the future as the king. Are there any legal or practical consequences for the British monarch if they make political comments and statements?
The statement was clearly a private one and not an official statement of the monarchy in any official capacity.
Members of the British royal family can of course have and express their own private political opinions (as anyone else can) if they like. They usually don't do that but of course they should not have less rights than anyone else. So they can.
As for the consequences, the most obvious would be that the British monarchy is somewhat dependent on government subsidies (the soveraign grant), so criticizing the government might risk financial retaliation. Critical comments can be seen as rather bold, brave, risky, careless or stupid, depending on the personal view on the matter.
Finally, in the long run such statements could either increase or decrease support for the monarchy in the UK.
In theory, it is still Her Majesty's government and if HM says jump, they are supposed to jump.
In practice, the UK has become used to an unelected head of state and an elected head of government, and careful courtesies between them. This is cloaked in concepts like the crown-in-parliament, and the fact that the monarch reads the PM's speech for each session of parliament rather than her own. So the saying goes that the crown will get at most one try at asserting written powers which may not be used under the unwritten rules. So the legal consequences are simply unpredictable. The UK is widely respected as a democracy.
The political consequences would depend on the issue, and on popular perceptions. Prince Charles has long held outspoken opinions on the environment and organic agriculture, which started out as mildly eccentric and which are now mainstream. As an example for the ruling monarch getting into political questions, think back to the Scottish independence referendum, which was as far as the current monarch was willing to push her neutrality.
For centuries now, the monarch has kept themselves from commenting publicly on practically any political issue. The last time an Act of Parliament did not receive Royal Assent was in 1708 and that particular instance was on the advice of Queen Anne's Government. While withholding Assent is one of the more drastic ways the monarch might influence politics, this timeline in and of itself shows hows rare the entire idea is nowadays.
Since Queen Anne, quite a lot has happened. A new Royal House was installed after her death as she left no heir; this imported Royal House became distinctly British with Queen Victoria (who was not eligible to become Queen of Hanover), the British Empire rose and fell, etc. What would happen if the monarch were to make a public political comment is a question of guesswork nowadays. As o.m. writes, they would get at most one try.
In all likelihood, we are not going to see an example that we can study. For one, the current monarch has and future monarchs will have staff to ensure that what they utter in public does not violate the unwritten principles. For two, the reigning monarch has far more subtle ways of influencing politics at their disposal. Most notably, there is the weekly meeting with the Prime Minister whose contents are understood to be entirely confidential (and to the best of my knowledge have been throughout Elizabeth's reign). I would expect the monarch to use such informal channels if needed and desired to state their stance on political issues to avoid their name being attached to any specific position.
The mileage varies for other members of the Royal Family who are not the reigning monarch.
