So, I have no idea whether this rumour is true or false. Given that, I'm going to weigh in with a view.
On the false side, this rumour is brought to you by the tabloid newspaper that passionately declared that Kris Marshall would replace Peter Capaldi as the next Doctor. Look how right they turned out to be about that.
If it really is true, then the principal thing is the scheduling of it all because Walsh has 2 gameshows that he hosts for ITV - they are "The Chase" and "Cash Trapped".
The Chase is one of ITV's most popular weekday shows at 3 million viewers per episode (yes, it's a quiz show at 5pm). It ran for 190 episodes last year, it runs for (possibly) 170 episodes this year (they're aired on 5 nights a week, but they take a couple of breaks per year.). The Chase film 12 episodes a week, which means Walsh spends a minimum of 4 months of the year filming that show. He won't quit The Chase to do something else. He loves it too much.
"Cash Trapped" is a quiz show devised by Walsh. Please don't ask me to explain it. Cash Trapped runs when The Chase isn't on and has been ITV's most popular Chase replacement since Tipping Point - that's another UK gameshow not involving Bradley Walsh (please don't ask me to explain that one, either). ITV are running the second series of Cash Trapped at the moment and, if this series is successful, they won't cancel it. Walsh won't quit that show either because he devised the show.
I should point out that both of these shows are for ITV, which is not the TV channel producing Dr Who. And who knows what the ITV contract says about Walsh working for the BBC, who are: "the other side"!
Also, Dr Who spends 7 - 9 months filming. I think that last year, they started filming at the end of July, and ran to April the next year (although it wasn't continuous and the cast got some time off in that (I think).)
So, either Bradley Walsh is going to be a very busy man for the next year, and the Dr Who filming fits around his other commitments, or the rumour's rubbish. On the busy side, I can't find any tickets for Bradley Walsh comedy shows for the next year apart from a panto around Christmas time. So, he's apparently not doing a comedy tour in 2018 - so far. That could free him up to do Dr Who. Equally, there haven't been any twitter denials of the rumour.
On a personal note, I would actually be rather happy if it did turn out to be true. Walsh appeared in the UK's version of Law & Order in a detective's role (show runner of the UK version of Law & Order was Chris Chibnall - so he knows Walsh, as he knows Whittaker from being show runner on Broadchurch.) and he was very good in it. I can totally see Walsh as a slightly washed out character with an ex-wife, a couple of kids that hate him and a gambling/alcohol problem being given a new view of life by the Doctor transporting him across time and space. (It probably won't be that, it's Saturday night TV on the BBC after all, but you can always hope.) If the casting rumour is true then, however Chibnall chooses to play it, their relationship has the opportunity to present a different take on the adventures in the modern series, which could turn out to be quite good.
I wait with a note of positivity about the whole thing.
