Interrogated every time crossing UK Border as citizen

I’ve never had cause to try this, but …

Short of employing a lawyer, you might make a data subject request under the Data Protection Act, for all personal data stored and consulted by a computer when you present your passport at the e-gate, and the reason for the subsequent delay. There are very limited grounds for refusal to provide access to one’s own personal information, and if they cite national security you will immediately know that you are on some list that you should not be on!

It will also cause somebody with managerial responsibility for the data to actually LOOK at what it says about you, and that might be all that is necessary to get the problem fixed. (Ie, they fix the data, send you a copy of the fixed data showing nothing untoward, and next time the e-gate magically lets you through! )

UK Border Control isn’t only about checking whether you are a British Citizen. The Officers have access to data of interest to other agencies, for example to the police. If you are wanted for arrest or interview by the police, you can be flagged on the Borderforce system, and Borderforce staff have powers to make you wait (for a defined and limited period of time, even if you are a UK citizen) till the police arrive and take over.

A problem that can occur is that the agency providing the information to Borderforce does not always have a full name or date of birth. Consequently the computer may flag up everyone of that name. If there’s a warrant out for the arrest of Paul Smith aged around 45, then anyone who loosely fits that profile may get flagged (and that’s probably why the e-gates won’t let you through).

The best you can do is raise this vigorously with the Home Office and ask them to ensure that you are flagged as NOT the person wanted on their system. Possibly to provide you with a letter to show to the Immigration Officer.

You can choose not to answer any questions when you enter, but I don’t think that will help you at all. They hold all the top cards at that point. Refusing to answer questions may seem odd and will probably guarantee a longer wait than if you co-operate.

"my passport is rejected by the electronic passport gates", it could be as simple as the RFID chip in the passport being damaged or corrupt. A polite note to the passport office with a description of the symptoms and asking them if this could be the problem could result in the passport being re-issued and your problems going away. If they insist there is nothing wrong with the physical passport, then a letter to your MP should help identify and unblock any issues with the Foreign Office.

As a UK citizen they have no legal basis to deny my entry,

Correct.

am I under any obligation to answer any of their questions

Yes. That’s their job after all. If they have reasonable grounds to believe that you are not a UK citizen, they need to do what they have to make sure you indeed ARE what you and your passport say you are.

and what can they do if I refuse to answer questions?

They can detain you until they have independently confirmed that you are a UK citizen.

How do I get this resolved, do I need to get a lawyer involved.

At this point a lawyer would indeed be the best option. Obviously there is something in the home office record that’s associated with your passport or person. Governmental systems are often arcane and convoluted and working with someone who knows how to deal with this can greatly increase the chances of success in finding out what it is and how to to fix it.

A lawyer can also advise you on what to say the next time you get flagged. There may be certain key words or trigger phrases to help things along.

This is indeed a frustrating experience especially if the underlying reason is really stupid. A friend of mine had a very common name, like "Paul Smith". Turns out there was a "Paul Smith" on an airline’s "do not fly" list and every time my friend checked he was questioned and bothered, despite him being a completely different person. He tried multiple times to have the airline straighten this out but they never did. Eventually he gave up and just stopped flying with this airline altogether.