The art of the thriller is to make the silly look plausible. Just think of every action film or television drama when the thoughts “hang on, she wouldn’t do that” or “come off it” float into your head and quickly dispel the dramatic suspense – as if a stray boom microphone had hovered into view.

It’s still more true of political thrillers, because we all know that politics is a business that revolves around people shoving emails around, shouting at each other and failing to make decisions. It’s mundane: to successfully make the characters and their lives larger and more vivid than their usually grey reality is quite the achievement.

On that basis (and many others, truth be told), Bodyguard is an extremely successful production. Imagine it as a cross between The Thick of It and a Bond movie and you’ll have the measure of it.

Or, thinking about it, a politicised version of the apparent short-lived affair between Princess Diana and her bodyguard, Barry Mannakee, which was ended when he was abruptly transferred and shortly afterwards died in a motorbike accident. You’ll not catch me spreading any implausible conspiracy theories, though.

The action in Bodyguard is twofold. There’s the usual thrillery stuff you’d expect about mass murdering terrorists trying to blow up trains, but the more emotionally engaging and thrilling dynamic exists between the two principal protagonists: tough, flawed and inscrutable close personal protection officer David Budd (Richard Madden); and the politician he is assigned to, the icy, ambitious and sexy home secretary Julia Montague (Keeley Hawes). Hawes plays her character like a more effective version of Theresa May, but with the enhanced sex appeal of, well, Keeley Hawes, I suppose.

Jed Mercurio’s excellent script can’t help but borrow from real life, and so here we have a May-esque stubborn, intelligent, slightly irritable woman tacking politically to the authoritarian right with “tough” policies and an eye to unseating her more moderate Conservative prime minister. As Ms May once yelled to Jeremy Corbyn at PMQs, “Remind you of anybody?”

There were other little vignettes from the political world too – an arrogant chief whip, for example, dismissing her bodyguard as “plod” in a nice echo of the “plebgate” scandal. The grind of the 24/7 media machine is also faithfully replicated, from the agenda-setting Today programme to the Andrew Marr Show (with Marr almost managing to play himself). Only a few racist gammons on Twitter were missing.

The treacherous and faithful special advisers – or spads – the red boxes and the ministerial Range Rovers deftly completed the dress-setting of this political stage.

I note in passing the programme advisers were Michael Prescott (longtime lobby correspondent) and Leigh Lewis (ex-senior civil servant who worked for, among others, David Blunkett and Iain Duncan Smith). There’s a certain authenticity to that side of things.

By the end of the first episode (of six) I was gripped by the slim, paranoid but unmistakeable possibility that Budd himself might be less of a bodyguard and more of an assassin.

Early on we learnt, in a sentence that was perhaps meant to catch the ear, and did, that he disliked politicians, when he shared some common humanity with a suicide bomber. He succeeded in talking her out of pressing the detonator on her suicide vest she had been brainwashed into wearing by her fanatical Islamist husband; Budd did so by reflecting that she and he were both victims of heartless politicians or ideologues sending them to their deaths but never having to risk spilling their own blood as they did so.

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Soon we see him playing a YouTube video of his new cabinet minister boss giving one of those parrot patronising interviews when they’re asked 11 times to apologise for something – in this case the Iraq War – and refuse to do so, Blair-style.

Soon we learn that Budd blames his time in Helmand for his PTSD, drink problem and the divorce he is going through, as well as the mess an old comrade of his found himself in, horribly scarred from some atrocity in the war against terror.

His friend from the Helmand days is now involved in a “peace movement”, and Budd makes clear that he’d still have a family and his mate would still have a face if the likes of Montague hadn’t packed them off to fight futile wars against Saddam Hussein and Islamic State.

We end up never quite sure whether Budd is ready to shag or to stab his political chief, and her intentions towards him are ambiguous too. Is the query “Is there a Mrs Budd?” routine small talk or a more serious interest?

Neither Budd nor Montague is especially likeable, but we care about these two and their developing relationship, and want to know which of the many turns it might take. I hope that this partnership doesn’t turn ridiculous with, say, some cliche sex scene.

Although the setup seems geared to some sort of physical relationship between Budd and Montague, Madden’s playing of Budd is so difficult to read that you start to think that it might not happen, which is dramatically the more interesting. The risk, though, with Madden making his character Budd quite so hard to read is that there ends up being nothing worth reading anyway.

Thus, as any good thriller should, Bodyguard starts your mind racing. Take the relationship between Montague and her spads. Was the male one (Paul Ready) as two-dimensional as he seemed? Was there more than sharing a Chinese meal between the two of them?

When Montague sacked her other spad, Chanel, a stereotypical millennial snowflake (Stephanie Hyam), Chanel turned very nasty indeed. She tried to sell the story of her “sociopath” boss to the tabloids but it was rejected (this bit wasn’t plausible, I have to say).

You got the impression that the slight but steely Chanel might, maybe, be set on a more violent revenge, possibly with another member of the PM’s entourage. And, if so, is bodyguard Budd going to pretend to be an accomplice in order to thwart their assassination plot; or was he going to join in?