Certainly if you use an online trading platform as your stockbroker then they make no mention of reporting gains to HMRC - and how could they, as you may be making losses on other platforms - so it's down to the individual to report through their tax return.
This article, "Ten ways HMRC can tell if you’re a tax cheat", is worth a read:
https://www.ft.com/content/0640f6ac-5ce9-11e7-9bc8-8055f264aa8b
Note it does not mention "reported by your stockbroker" but it does mention Suspicious activity reports:
9. Suspicious activity reports
If a bank or professional adviser has any suspicion that a customer might be involved in money laundering or terrorist financing, they are obliged to alert law enforcement agencies. Every year, hundreds of thousands of “suspicious activity reports” (Sars) are filed. Nine out of 10 are filed by banks or other financial institutions, with the remainder from tax advisers, accountants, lawyers, estate agents and others. Since November 2013, Sars data has been transferred to HMRC’s Connect database on a monthly basis.
Tax evasion is an offence under the money laundering rules, so it is no surprise that HMRC pays close attention to these reports. Jonathan Fisher, a tax barrister, says it has been estimated that one in four HMRC investigation cases were triggered by Sars. He says: “Sars are definitely a key provider of information to HMRC regarding suspected tax evasion.”