In a recent episode of Poirot, Inspector Japp said that he'd been working for Scotland Yard for more than thirty years and had never seen a case quite like this. Later on in the same episode as Poirot, Hastings and Japp were looking on as a coffin was being lowered into a grave, I noticed that the date of death of this poor soul was 1936.
Assuming that Japp joined Scotland Yard exactly thirty years and one day previous, then the latest date which he could have joined the Yard was 1906.
We know from the Sherlock Holmes canon that Inspector Lestrade was still at Scotland Yard as late as 1902 but the books were still being written as late as 1924. This at least hints at the possibility that Japp and Lestrade were both working for Scotland Yard at the same time.
I wonder, would Japp and Lestrade have ever come across each other? Maybe. If that is true then I would expect that not only do Japp and Lestrade know each other but I suspect that Japp would have worked as Lestrade's offsider. Japp could have logically been Lestrade's apprentice. This would explain in part why the Yard thought it useful that he was the one who ends up tailing Poirot so often, having previously done so with Holmes.
There is a major problem though. Scotland Yard (as a metonym) is the headquarters of the Metropolitan Police Service which serves all of central London. By 1890, which is bang in the middle of the Sherlock Holmes canon, the Met had 13,000 officers and was such a large organisation that it had to move to New Scotland Yard; which are now the called the Norman Shaw Buildings on the Victoria Embankment and later housed among other things, the offices of the Offices of the Leader of the Opposition. It is quite possible that David Cameron and Ed Miliband have worked in an office that either Japp or Lestrade could have worked albeit incredibly unlikely.
Functionally Japp serves an identical purpose to Lestrade in the novels and even goes through the same process of metamorphosis of extreme distrust to gradual acceptance of their respective detectives.
Forget trying to publish a story with your hypothesis though because although Sherlock Holmes logically entered the public domain for copyright purposes in 2000, Poirot will not until 2046. The stories of Japp and Lestrade would by that stage, very much be historical fiction and well beyond the memories of anyone alive.
The question of whether or not Inspectors Japp and Lestrade could have met remains interesting but I fear that not even Holmes would be allowed to publish a monograph on the subject for a very long time.