In England and Wales, in the years from 2004 to 2008 there were between 83 and 155 prosecutions for perjury each year (about 2-3 per million people), with a high (but not 100%) conviction rate.
(Freedom of Information Request Source)
Note, however, that not all (or even most) prosecutions for perjury involve in court testimony. Out of court statements made under penalty of perjury or under oath (such as in affidavits) can also be a basis of a perjury prosecution.
For comparison's sake with the information on Colorado below: "On Census Day, 21 March 2021, the size of the usual resident population in England and Wales was 59,597,542 (56,490,048 in England and 3,107,494 in Wales)" which is about 12 times the population of Colorado, which has roughly the same per capita rate of perjury prosecutions.
How often does it actually happen that a party is actually proceeded
against for perjury / misleading a judicial body / contempt of court,
or faces any other consequences for doing so other than losing their
dispute/point/case? And whatever the answer, why is this?
Perjury prosecutions based upon court testimony do occur and result in convictions, but they are very rare.
Colorado is a state with a population of roughly five million people. There are many hundreds of thousands of court cases in the state each year in which often multiple affidavits or declarations are filed and depositions are taken under oath, there are thousands of evidentiary hearings in the state courts of the state each year, and there are myriad documents executed by its citizens under oath or under penalty of perjury outside a court setting (e.g. in government paperwork).
An intentional false statement of material fact (i.e. a lie) made in testimony made under oath in a hearing, trial or deposition, or in an affidavit or a declaration or other document signed under penalty of perjury, can be prosecuted criminally in a perjury case under state law.
Most of the statements made under oath do not contain perjury, but a significant minority percentage of such statements do. Even if perjury were committed just 1% of the time (which is probably low), there would be thousands of instances of perjury committed each year in the State of Colorado.
How many perjury prosecutions were commenced in the entire state of Colorado in all of its state courts (except Denver county court which is statistically separate) in the 2019 fiscal year (ending June 30, 2019)?
There were 5 filed in District Court, which were probably felonies, not all of which would have involved in court testimony.
There were 2 filed in juvenile delinquency actions (and most likely did not involve in court testimony).
There were 6 filed in County Court, which were either misdemeanors or were criminal complaints that would later have been transferred to District Court and would be in that case duplicative. These cases are also unlikely to have involved in court testimony.
A significant share of those prosecutions, moreover, would be for instances of perjury not involving testimony in court. Probably only 1-4 cases in Colorado involving court testimony under oath are prosecuted in a given year out of thousands of cases each year where perjury actually happens. It would be rare for more than one to go to trial each year and in many years, no cases of this kind would go to trial.
For example, some of those prosecutions would involve a false statement made under penalty of perjury in an application or report to an administrative agency.
There is no reason to think that plea bargaining rates are significantly different in perjury cases relative to other kinds of criminal cases, so there are probably no more than one or two perjury trials in the entire state of Colorado in a typical year.
Despite the fact that the threat of a criminal prosecution for perjury is a backbone of our court system and many of our administrative and civil legal processes outside of court, perjury prosecutions are almost never brought.
Prosecution practices may differ from state to state, but Colorado is almost always statistically typical of the U.S. on almost every state by state statistic (except obesity, Colorado is the thinnest state). Colorado's pattern of perjury prosecutions is the norm. The threat of criminal prosecution for lying under oath is ever present, but the reality of such a criminal prosecution is almost non-existent.
Contempt of court proceedings brought in the same criminal case or civil case in which the perjury took place are possible and probably more common than perjury prosecutions (and have basically a misdemeanor offense punishment in this context) but are still very rare.
One of the reasons that this is rare is epistemological.
Proving that someone made a false statement of a material fact is relatively easy and is sufficient to get a correct result in the matter being decided by the court. But proving that someone made a false statement intentionally, rather than due to a mistake of memory or perception is much more difficult to establish beyond a reasonable doubt. Even in cases where a judge makes a finding of fact that a particular statement upon which a witness testified is not credible, judges typically go to great pains to refrain from accusing the witness of actually lying unless no other explanation is possible.
Another factor is that when a party to a case lies under oath and the judge or jury doesn't believe that party's testimony, the negative result in the lawsuit or criminal case that results from that determination is often considered a sufficient punishment so the extra burden of a collateral perjury prosecution is not justified.