I do not know if there are any records for Osborne, London serial number.
Vol 2 on Nigel Brown’s British Gunmakers p123 for a Charles Osborne, Birmingham, would put 78xxx between 1923 & 1925. That is if the 78xxx was the gun serial number. As for the 3 digit frame serial number not matching the barrels it appears to indicate a barrel replacement by Osborne, not necessarily a complete gun by them.
Is there a serial number on the forend ? If so what is it?
Are the barrels only engraved "New barrels by Charles Osborne & Co. London" or do they have an address? An address would help date the barrels as the London location moved a round a bit.