Hi Alex,
As said above, all of the joints must be tight. All big Healey steering boxes were worm & peg.
Additionally, there is an adjustment for peg end play, located on the top of the steering box. Also the steering shaft (from steering wheel) has a shim adjustment for end play. Shims are removed from the steering box end cover. With things correctly adjusted, there is usually no free play or binding.
In rare cases, if the worm gear is too worn, a replacement worm/shaft is available.
Denis Welch, UK has a "constant clearance worm" available which removes any "off center" play, but it is quite expensive. The original design allowed for slightly more play when the steering is off center.
I don't know of an easy replacement rack & pinion conversion, but someone else may know of one.
D