Firstly, in terms of contractual and/or tortious liabilities, the law in relation to limitation is notoriously complex and anyone thinking about how to deal with retention of paperwork should certainly talk to their liability insurers for advice.  Even where the limitation period in relation to a particular contractual or tortious liability is 6 years (which is often quoted as the legally required time), there may be debate about when "time starts to run" and in practical terms the court documents in relation to such a claim do not have to be served on the defendant immediately they are issued at court, so to allow for some leeway, a minimum period of 7 years might be wise, but as mentioned above, it would be best to take specific advice from your insurers.  The situation can become very complex if for example the matter has involved a child, a protected party or a claim for provisional damages.
 
If  we are interpreting your question correctly then you are not asking about keeping documents to satisfy HMRC or to rely on in the event of litigation against your firm, Rather we think  you are asking about the need to keep records in case the matter to which they relate goes to a trial in court and your (your expert's) evidence is needed.  The law of limitation does of course not govern this time period.  
 
The ideal approach in this case would be to: "when the case settles, then your system should automatically delete all info, other the invoice and the actual report."  
 
That approach is, of course, dependent on the solicitor telling you that the case has settled.  
 
The difficulty for you might be to know when a particular case has concluded.  Some cases may well take more than 6 years from the date of instruction to conclude and it is conceivable that some of those cases could go to court and thus you would need to have access to the relevant documents.  The instructing solicitor might well be able to provide copies of many of the relevant documents but the expert/your firm would be expected to have kept any records of their own until the conclusion of the case.
 
You will also have to consider your GDPR position and be aware of not keeping personal data for longer than is necessary.
So on a practical level, you would ideally retain all records until notified that a case has concluded and then for the appropriate period thereafter as describe above in terms of contractual liabilities.  
 
Touching on the point of whether to keep paper or electronic records, we would add that if you elect to keep the records electronically you need to be mindful of security and back up.
 
In summary, as a minimum:
 
1.     From a liability perspective, you should talk to your insurers ;
2.     From an on-going duty to the client perspective, you should talk to the instructing solicitors to clarify their expectations and to agree a policy;

There is some helpful advice on this page including an example saying that an organisation can keep limited amounts of data available about previous clients if there is a possibility of a complaint:

Principle (e): Storage limitation | ICO