UK Family Visitor Visa refused on account of having a criminal conviction, however I've never actually had one. What are my options?

It's about deception. You applied for entry clearance and were refused on Paragraph 3.6a of Appendix V...

V 3.6 An application will be refused where: (a) false representations have been made or false documents or information have been submitted (whether or not material to the application, and whether or not to the applicant’s knowledge); or

The grounds were that you did not disclose a prior criminal conviction. But you believe a mistake has been made. This is going to be messy.

Background. V 3.6a is a plain English summary of Paragraph 320 (7A), which says...

(7A) where false representations have been made or false documents or information have been submitted (whether or not material to the application, and whether or not to the applicant’s knowledge), or material facts have not been disclosed, in relation to the application or in order to obtain documents from the Secretary of State or a third party required in support of the application.

More background. The ECO uses the civil standard of proof for everything in an application, 'balance of probabilities'. BUT the UK courts have ruled decisively that when there's a refusal under V 3.6 (or 320 7A), the ECO must be held to the criminal standard, 'beyond reasonable doubt'. And the courts have placed the onus of proof on the ECO. Plus this type of refusal triggers an automatic review by the ECM. And depending on the consulate, the refusal may be flagged up for the Chief Inspector.

So when there's a refusal on deception, we already know that the ECO is really really sure about it. It's not a rumour, it's not the 'balance of probabilities', it's not a mix-up of names, it's not a mistaken identity, it's not a wild stab in the dark, we know that the ECO is prepared to acquit himself of his onus of proof, and he can do this 'beyond reasonable doubt'. And a senior official has agreed the findings. All of this happens before the refusal notice is even drafted.

What this amalgamates to is they found something on you that "stuck". They took the step of speaking with you on the phone. That was your chance to avoid the refusal. Note that when an ECO calls somebody on the phone it's an extraordinary event and you need to tread very, very carefully. It is indeed NOT a conversation.

In the very worst case, they may have got a cold hit on your biometrics.

Using the help of a solicitor I've applied to some Crime Record Bureau to check for records. But I've been told it will take 40 days. I am filling for SAR from UKVI as I swear I do not remember ever being convicted.

What are my options to fix this situation?

First, these things are messy and will take a long time. It means nothing is going to be fixed in 40 days. If you had travel plans, you can put them off indefinitely.

Next, it seems you do not know where the conviction occurred, but any place that you have been is a candidate. It might be a bench warrant in the US for example which you would naturally be unaware of. It might have occurred on a prior trip to the UK, a CCJ that got communicated back to your authorities, another zinger is a default on child support resulting in a contempt of court judgement. Those are examples of things where the person may be unaware. So you'll need to instruct legal counsel to start looking. Start locally and expand your search incrementally to any place you have been.

Next, is the Subject Access Request, you have started this and need to follow through on it. They will not release the exact information they found because they will classify it (and they have the right to do that). So your SAR result is likely to be dry, and you will need to appeal it to the Information Commissioner, perhaps several rounds of appeal on your SAR alone will be needed.

If after all of that you don't find anything, you'll have to make a fresh application and answer 'yes' you appear to have a conviction because you were refused on V 3.6a, but do not know what it is or where it happened. Include all the evidence of how you searched for something. In doing that, they can't get you for deception again. They can still refuse, but at least their grounds will not be something that attracts an outright ban.

Finally, if you are convinced that you have a clean history you can take the position that their decision is perverse. That route takes a lot of time and money (and risk) and there's no point in repeating it here because there's an adequate road map here What can I do to lift a 10 year ban for making a mistake in my UK visa application? in the description of judicial review. This step is a last-ditch 'hail mary' and I recommend avoiding it.

Concluding on your comment...

I am impressed with Mr. Gayot Fow. Can I have conta ct details of Mr. Gayot?

Gayot doesn't take casework and will never get involved in something where a person can be uniquely identified as a particular individual. Not even in private and not even for lots of money. His involvement on this site is purely recreational, and it will stay that way.