Generally speaking, jury trials in England and Wales are allowed (but can be waived by a defendant in most cases) for offenses punishable by more than six months of incarceration (i.e. serious misdemeanors and all felonies), with exceptions for domestic violence cases, serious and complex fraud cases, Diplock courts in Northern Ireland prior to 2008, cases where there is a serious risk of jury tampering, and in cases where double jeopardy defenses are at issue. It appears that jury trials are mandatory, however, and cannot be waived by defendants, in some serious criminal cases. (Historically, trial by jury was suspended in most cases during World War II.)
English law enforcement officials can also detain people without trial for limited periods of time (generally up to 28 days prior to 2011 and now 14 days) without criminal charges in terrorism cases. This detention without trial procedure was instituted at the same time that the Diplock court system was shut down. (Courts-martial present another issue entirely.)
Diplock courts were instituted out of concerns about jury nullification and jury tampering, and the latter concern remains an all purpose reason to deny a jury trial in a particular criminal case.
Importantly, since 1967, juries in England and Wales need not be unanimous. Up to 2 dissenting votes are allowed on a jury of 12 and up to 1 dissenting vote is allowed on juries with 8 to 11 members remaining at the time of deliberations. The failure of up to three jurors on a jury of twelve (used in the Crown and High Courts) to complete the trial is allowed without a substitute and up to 1 juror who doesn't complete a trial is allowed in a jury of eight in County Court.
From the first link above:
A coroner must summon a jury for an inquest if the death occurred in
prison or in police custody, or in the execution of a police officer's
duty, or if it falls under the Health and Safety at Work etc. Act
1974, or if it affects public health or safety.
Coroner's juries have 7-11 members and a minority report of 2 members of that jury can be its verdict.
In civil cases, juries are used in eminent domain valuation cases, where the judge might be seen as having a conflict of interest. Further, as explained in the first link above, they are also used as follows:
In 1998 less than 1% of civil trials in England and Wales were jury
trials and these were principally defamation cases.
Section 69 of the Senior Courts Act 1981, which replaced s. 6 of the
1933 Act in respect of High Court trials, provides that trial shall be
by jury on the application of a party where the court is satisfied
that there is in issue:
a claim of fraud against the party; or
a claim in respect of libel, slander, malicious prosecution or false imprisonment
unless the court is of the opinion that the trial requires any
prolonged examination of documents or accounts or any scientific or
local investigation which cannot conveniently be made with a jury.
FYI, Scottish practice with regard to jury trials is completely distinct from the practice in England, Wales and Northern Ireland and is beyond the scope of this answer. Both the size and process of the jury, and its availability are different in Scotland.