There'd be a part of me that isn't always sad whenever I watch Dad's Army. Beck's portrayal of Walker was charming, nuanced and genuinely one of the standouts of the show. He was a great comic actor and I've no doubt that had he not had his health problems he'd've gone on to be a stalwart of the British comedy on stage and screen.
I don't think it would change the overall story arc of Dad's Army, though, as the show never really had one. You'd get a few more Walker-centric episodes, for sure, and he'd have some memorable scenes getting things together for Jones' wedding in 'Never Too Old' (the Finale) but I don't see a massive change.
As for his own career, I don't know enough to guess where he'd end up. Ian Lavender (the only other cast member close to his age) had a relatively modest career on TV but I believe remained a strong stage actor and appeared on EastEnders in the early 2000s. Beck would have been a recognisable figure and could have done similar, or for all we know might have retired to a quiet life of royalty checks, chat shows and memoirs.
Either way, the only thing we can say for sure is that Dad's Army would have been that little bit closer to perfection. RIP James Beck, we lost you too soon.