According to the BBC Pronouncing Dictionary of British Names (1990), the Christian name (forename) Leveson is ˈluːs(ə)n, and the surname Leveson-Gower, the family name of the Earls of Granville, is ˈluːs(ə)n ˈɡɔː(r).
I suspect that as a forename Leveson is now obsolete, and I don't believe there is any Leveson-Gower now active in British public life, though several members of the family were active in politics in former centuries. Furthermore, Sir H.D.G. Leveson-Gower was a well-known cricketer, captaining the English test side in 1909/10. His brother Frederick was also a county cricketer, and in 1895 played for the Gentlemen of England against Oxford and Cambridge. (Oh, long-lost era of gentleman amateurs!)
At one end stocky Jessop frowned,
The human catapult
Who wrecks the roofs of distant towns
When set in his assault.
His mate was that perplexing man
We know as "Looshun-Gore",
It isn’t spelt at all that way,
We don’t know what it’s for.
But as with Cholmondeley and St. John
The alphabet is mixed,
And Yankees cannot help but ask -
"Why don't you get it fixed?"
EPD/CPD confirms ˈluːs(ə)n in Leveson-Gower, but does not tell us anything about a forename. It does, however, say that the simple surname Leveson is ˈlevɪs(ə)n.
Back to Sir Brian. He does not appear to be related to the Leveson-Gowers. The BBC Pronunciation Unit tell me that some fifteen years ago his office gave the pronunciation of his name as ˈlevəsən, which we can take as a minor variant of ˈlevɪs(ə)n. Jo Kim tells me “I checked this again with the Sentencing Council press office today (Lord Justice Leveson is the Chairman) and they confirmed that ˈlevəsən reflects his own pronunciation“. (Thanks, Jo.)
That didn’t stop the Home Secretary, Theresa May, referring to him as Lord ˈliːvɪsən in the House of Commons yesterday.