The 128 was a great little car. I loved them and I worked on them (can't say that about too many cars I worked on.....
The original tyre for the 128 Saloon/Sedan was 145 R 13, often Pirelli Cinturato, iirc later they used Michelin ZX or possibly XZX.
www.longstonetyres.co.uk has a guide to which tyre is used on older/classic cars, including the available modern equivalent.
E.g. for the 128, they recommend and stock the Pirelli Cinturato CA67 in 145R13. They describe these tyres as being tubeless, but iirc the 128 used radial tyres but they were tube type. Unless there are safety humps on the wheel rims (I can't tell from your picture), tubeless tyres should not be used without tubes. They have these tyres/tires in stock at £340 for 4 excl. vat.
As you're in N.Z.. you presumably want to source your tyres, there? but the above website is useful to consult for application info. This particular tyre size was also used on the MG Midget, Austin-Healey Sprite, Triumph Spitfire and Lotus Elan Mk1. These other vehicles applications might help you find a tyre supplier, perhaps there's a classic British specialist near you that stocks tyres.
Afaik, if you go down one step in aspect ratio you can go up one step in width, so it would appear that 155/70R13 would be suitable to replace the original 145R13, with virtually the same rolling radius and speedo readings.
You could replace your existing tyres with another set of 155/70R13. Be aware if you go wider than 145, your spare tyre or one removed from the vehicle after e.g. a puncture, might not fit in it's normal place under the bonnet, sorry hood.
Re:- you indicators not self-cancelling. There's a small spring loaded plastic pin on the underneath the steering wheel (you'll have to remove the steering wheel to see it) that is supposed to engage in a hole in the plastic ring that sits above the indicator switch on the steering column.
There is a certain amount of adjustment up and down on the steering column switch assembly - this adjustment is accessible through a hole lower down on the column cowling/plastic shroud behind a large blanking plug, if you look in you should see a clamp with an 8mm bolt head, loosen this to allow the column switch assembly to be adjusted. If you can adjust the column switch up, do so, then lock it in position, turn the steering wheel both ways to allow the spring loaded pin to engage the indicator switch ring, described above.
It's possible that the pin has been broken off the rear of the steering wheel or that something has broken within the self-cancelling mechanism of the indicator switch.
So if the above adjustment isn't successful, you'll need to remove the steering wheel. (It's not difficult). Prise the horn push button off using a knife or small screwdriver. Underneath remove the large horn-push spring. Loosen the 24mm nut a few turns but don't remove it yet. To loosen the steering wheel on it's splines - and this is a special tip, I invented it
, - grip the steering wheel firmly on the left and right of the wheel adjacent to the wheel spokes, now push down on the left side and pull up on the right side, then reverse the action - as if you were trying to rock the steering wheel on it's splines (you are!), sooner or later the wheel will come free, remove the nut and then the wheel. Turn over the wheel, you should see the little spring loaded pin (if it's still intact). If you look at the top of the steering column switch assembly you'll see the plastic indicator-cancelling ring that I mentioned above. If you operate the indicators left or right and then you rotate this plastic ring, you'll operate the self-cancelling mechanism and be able to see how it works.
Only thing to watch for is not to damage/bend the 2 brass contacts for the horn push button on the steering column (easy to do so). When you refit the steering wheel you may have to rotate the st. wheel left and right a bit to allow the pin to engage with the hole in the rotating ring.
Hth,
Al.