What if I get stopped by immigration officers and I have no ID on me? I am a British citizen.
There is no general duty to carry your identity with you when inside the UK, nor to identify yourself to any official. Many UK citizens have no photographic proof of identity at all. (My parents did not for many years, until they obtained fresh passports.)
I am confused where you could be stopped by immigration officers. If you appear at a UK Border without your passport, it is a real headache, but immigration can find your record on the computer and will, if you satisfy them you are a UK citizen, eventually admit you. (They have no power to deny entry to a British Citizen, and must be satisfied you are not a British citizen to deny you entry.) For clarity, in respect of some comments, I am not advising that anyone should do this, but I am saying in the worse case scenario if somehow you end up without documents you can still be looked up in the computer, and your identity can be checked at the UK Border without a passport.
In general you are not required to identify yourself to a police officer, unless you are arrested. http://www.findlaw.co.uk/law/criminal/your_rights/500109.html
If I am stopped and searched, do I have to give my name and address?
Although the police will likely ask for your name and address, you are not required to give it unless the police arrest you or are reporting you for an offence.
There are a small number of occasions when you may be required (by law) to identify yourself when you are simply going about your private business. If you are driving a vehicle the police may stop you and require you to identify yourself. If you do not have your driving licence, you may produce it at court at a later date; in the mean time the police can check your status by computer. http://www.legislation.gov.uk/ukpga/1988/52/part/VII/crossheading/powers-of-constables-and-other-authorised-persons/enacted?view=plain
If you are travelling domestically inside the UK by air, the police can stop you and require photographic identification at or after the security check point under Schedule 7 of the Terrorism Act 2000. (http://www.legislation.gov.uk/ukpga/2000/11/schedule/7?view=plain) However, I travel within Great Britain by air several times a month and usually do not bring anything more than a credit card and a change of clothes, and I have never had any problems. Most airlines will "advise" you to bring photographic ID.
If however you do travel between Great Britain and Northern Ireland, immigration staff are often interested in your status then. It is wise to have photographic ID on such a journey.
I've yet to find any source for this, but no: to the best of my knowledge, there's no requirement to carry any form of ID at all times in the UK. It's purely used for specific situations, e.g. buying alcohol. I certainly don't carry my passport on me at all times, and I don't think I know anyone else who does. There certainly seems to be no statute requiring British citizens to carry ID.
Should you encounter a situation where identification of some kind is required - speeding when you don't have a copy of your driver's licence to hand is the most common example - you have 7 days to show your documentation to the police.