It is not usual to call people with an honorary doctorate "Doctor" except in the context of the awarding institution. Most likely the awarding institutions will have actually specified that the recipient should not give anybody the false impression and I can't imagine Tony is the type to do otherwise.
His title at Oxford was 'Professor', and he was addressed as 'Tony'.
He made incoming DPhil (PhD) students a cup of tea individually in his office at the Computing Laboratory. It was a small group, but still I appreciated this personal touch.
I never met Tony, but I liked his work. I'm not much of a one for tea, but I don't think either of my PhD supervisors ever bought me a drink - I didn't finish (got cancer, I'm fine now†, some cancers are very curable, but frankly I was struggling anyway so it was a good excuse to quit) and I'm sure it's traditional to buy something a bit harder than a cup of tea if you pass, but I didn't get that far.
Anyway my point here was just a PSA that honorary degrees "don't count". If somebody only has an honorary doctorate but insists on being called "Doctor" they're an asshole. In fact, even outside University I know a lot of MDs and PhDs and in most contexts if they insist on the title "Doctor" they're an asshole even though they're entitled.
† Well not fine, I'm old but I think that's an inevitable side effect of surviving so the alternative was worse.
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