Usually, for a road bike – tourer or sports bike – not a race bike, the tyre is 3mm wider than the brake track width. Eg 20mm brake track, 23mm tyre. 22mm brake track, 25mm tyre. 25mm Brake track, 28mm tyre.
On a race bike, 18mm brake track, 20mm tyre ( 18mm rims will be triple wall and the tyre will be inflated to 150 psi. )
The brake block should be set 1mm clearance from the rim, so to avoid little scuffs on the tyre wall, quick release mechanisms were introduced on brake callipers. With non QR brakes, the tyre can be 'thumped' through.
On a race bike, a 20mm tyre will slip straight in without a QR, thus providing a speedy wheel change.
A 23mm tyre is the narrowest I would dare on British roads.