I've been looking into this some more and found some questions along the same lines (unfortunately without useful answers) on bitcoin.SE as well. However, I also found this publication from HMRC which is actually a very useful overview of tax treatment of cryptoassets.
The whole document is useful and relevant but I would particularly point you at the following section (my emphasis):
Individuals need to calculate their gain or loss when they dispose of their cryptoassets to find out whether they need to pay Capital Gains Tax. A ‘disposal’ is a broad concept and includes:
- selling cryptoassets for money
- exchanging cryptoassets for a different type of cryptoasset
- using cryptoassets to pay for goods or services
- giving away cryptoassets to another person
If cryptoassets are given away to another person who is not a spouse or civil partner, the individual must work out the pound sterling value of what has been given away. For Capital Gains Tax purposes the individual is treated as having received that amount of pound sterling even if they did not actually receive anything.
In other words - yes, you are liable for CGT (if your gains are above the threshold) and you will have to document every trade. If you genuinely don't have records to support each trade then I would call HMRC for advice about how to proceed. They are usually very helpful on their phone lines and will keep escalating you upwards until you find someone who can answer.