John Sessions is drinking coffee and making a call on a prehistoric mobile phone when I arrive at the Soho hotel where we are having lunch. Sessions holds it up proudly. "I don't have an iPhone, I don't do apps," he exclaims. At 61, he has embraced bufferdom. Grey-haired and wearing large spectacles, he looks like Ronnie Barker, or a retired colonel you might run into at a golf club in Surrey.

He admits it, rejoices in it. "I rang the BBC the other day," he tells me. "I'm sick to the back teeth of hearing people say 'mitigate against'. 'Militate' is the verb. Martha Kearney did it. I said: 'Will you please tell Ms Kearney, it's militate.'" He votes Ukip, hates the "commissars" of Europe and, naturally, lives in Wimbledon. In his new film, Pudsey the Dog: The Movie, he plays Basil Thorne, a deranged version of this buffer who tries to eviscerate both Pudsey – the dancing canine that won Britain's Got Talent in 2012 – and a blissful local village that Thorne believes would benefit from a shopping mall.

I suggest to Sessions – ace impersonator, voice of many a Spitting Image puppet, consummate actor, king of improv – that bufferdom is just another persona for him to hide behind. "Oh, you mean the man of a thousand characters," he says doubtfully. "I'm pretty much one character really. A grumpy old fool."

The film, a sort of Lassie meets The Railway Children, is terrible, although clearly I am not in the four- to 10-year-old target audience and have limited interest in talking horses and farting pigs. Why do it? "I got offered the job and thought, 'Children's film villain – that could be quite jolly.' I tried to give it a bit of a Dennis Price sneery thing." But what about WC Fields's golden rule that you should never work with children or animals? "In my experience," says Sessions, "it's not the children that are the problem, it's the parents."

Shouldn't he be giving his Richard III – a role he's talked about performing in the past – rather than playing comic-book villains in forgettable kids' films? "Far more distinguished actors than I have done children's films and then returned to putting on the tights and crown," he says. But playing second fiddle to a dog? "If that's the worst that happens to me," he says, "I'll be very lucky."

Everything seemed possible when Sessions was riding high in the late 1980s and early 90s: Whose Line Is It Anyway? and Porterhouse Blue on TV, My Night With Reg at the Royal Court and in the West End. But stardom never quite materialised. As the starters arrive, I ask Sessions what happened. "I had a twinkly couple of years, but then I ran out of steam," he says. "I turned 40. As I was getting older, I wasn't getting more confident, I was getting less confident. I lost my way."

Did he have a career plan? "When I left Rada [in the early 80s], my plan was to try and do two careers at once – to be a comedian and an actor. For some years, I managed to juggle the two, but I never felt I joined either club. I still feel like a rookie." Actors saw him as a comedian, comedians thought he was an actor. Falling between two stools affected him. "I feel inadequate when I'm working with great actors like Michael Gambon. It's like talking to God."

While appearing in My Night With Reg in 1994, Sessions had a memory lapse during one performance and had to briefly leave the stage to gather himself. This left a legacy of stage fright, and he did not return to the theatre until last year. He realises now that being away from the stage, which he calls the "engine room of the career", was damaging. "On stage, you have to stand up and be counted. In film and television, the editor can save your arse in so many ways."

He returned last year in William Boyd's Longing at Hampstead theatre. "I was very workmanlike and learned the whole thing off by heart before the first rehearsal. It was a terrific cast and went very well." It has not, though, produced a spate of other offers. "I thought it was going to lead to all kinds of interesting things, but I wasn't killed in the rush."

His stage fright came at a watershed in his life. His mother had just died, he was on the point of coming out as gay, and he had just passed 40 – he has a streak of melancholy and frets about age and mortality. He realises now what he should have done. "After My Night With Reg, I should have gone to the RSC or the National and done four or five plays, really worked my arse off. Some good old-fashioned graft would have done me the power of good. But after a six-month run, I couldn't face a play again."

So how does he see the arc of his career? "I wish I could accord it something as grand as an arc," he says wryly. Stella Street, the BBC2 series he co-wrote and starred in, is perhaps the best of his later work. It imagines a street in Surbiton occupied by celebs, including Michael Caine, Dirk Bogarde, Jack Nicholson and Joe Pesci. The cornershop is run by Mick Jagger and Keith Richards, and Jimmy Hill pops up occasionally to bore everyone rigid. It was funny, odd and became a cult hit, which means it never quite got the ratings it deserved. A later film based on it fared similarly, winning awards in the US but no audience.

Does the dimming of the twinkly years and the failure to play Richard III bother him? "I beat myself up about it," he says. "Life's not a rehearsal and all that. But you can't change it now." It's hard to know how deep the wounds really are, and throughout our conversation he is keener to talk about the talents of others – notably the late Rik Mayall – than his own. "We were good friends," says Sessions. "We were very different people, but that's how good friends work. I was first with him – but too scared to speak to him – when he was doing the Comedy Store in the 1980s and I was doing rarefied shows about this, that and the other." One such show imagined Milan Kundera's version of Dallas, a quintessential Sessions idea jolting together two antithetical genres, the shtick that was so successful on Whose Line Is It Anyway?

Sessions is easy to talk to, thoughtful, honest, willing to reflect on his disasters as well as his successes, less self-possessed than when he assumes a character. But the surface bluffness and the hints of inner turmoil are hard to square. He mentions being bullied at school, has had periods of depression, and was in therapy for a long time, but when I ask him about this he is keen to downplay it. "I'm hung up to the extent that most people are," he says. "We all have urban angst." Oh come on, there must be more to it than that. Pressed, he accepts there was, and that he did suffer depressive episodes. "It was treated with anti-depressants, and I hope I'm through it," he says. "I feel all right now." He says the triggers were worrying about whether he was going to get enough work and feeling that he was a failure – not least as his friends and contemporaries garnered knighthoods and other awards. But in the past four or five years, he has found a new equanimity, helped by the fact that he is "nicely busy", not least playing Humpty Dumpty alongside Johnny Depp in a forthcoming film of Alice Through the Looking-Glass.

He reckons that because interviewers invariably ask him about depression and the fact that he's gay, people must think he's always banging on about it – hence perhaps his reluctance to bang on about it. In one previous interview, he had quoted Yeats's poem Never Give All the Heart (he is a great citer of lines from poetry and plays). Did he ever give all his heart? Yes, he says, sort of, but to two men who were happily married – old friends who remain close to him.

Later, when we are talking about two other close friends, Kenneth Branagh and Stephen Fry, he gives a better answer to the question of why his career stalled. "I lack the edge that makes Ken what he is and makes Stephen what he is. I have a great feeling of inadequacy, and lack the spur to prick the side of my intent." He is uneasy quoting Macbeth and has slightly garbled the line, but you get the gist. "I was a hungrier man as a younger man," he adds. As if to prove the point, he leaves most of his lamb rillettes.

Towards the end of the meal, he tells me he is making notes for a new one-man show with the working title Nutty Professors, about the generation of mad old men – such as FR Leavis and G Wilson Knight – who dominated English literary scholarship when Sessions was studying for his degree 40 years ago (he did an MA at Bangor and spent four dismal years in Canada studying for a PhD on the English poet and novelist John Cowper Powys). "They are all hilarious, and what is most hilarious is that they sit down and write books about the blossoming of the human spirit – while having a venomous fight with some guy in the English department over where the coffee machine should be."

The proposed show echoes a whodunnit based around the Bloomsbury group that he once planned to write – another collision of moralising art and messy life that combined his comic sensibility and love of Eng Lit. The novel, like the PhD, was never completed, but let's hope the show does see the light of day. Sessions is too clever and talented to be a support act for a dancing dog.

• Pudsey the Dog: The Movie is on general release on 18 July.