It depends on a lot of things, including how far you are travelling, whether you have support, how quickly you intend to travel, and which bike you actually have for example. With the trend for wider tyres and gravel bikes, that there are much better options for road bike tyres in larger sizes. A 32mm road tyre like the Continental Grand Sport Race should fit your frame, and be significantly quicker than knobbly CX tyres.
When I rode from Dungeness to Thurso last year I used my touring bike, which has nice low gears, a comfortable steel frame and a slightly more upright position. However I was carrying some luggage as I was unsupported, and even though I was using hotels I maxed out at 100 miles a day.
For that sort of tour, a gravel-bike style CX would be not to dissimilar to my tourer. I actually also have a Genesis Vapour now which is supposedly a CX bike but is rather more relaxed than a full racer, and I would be perfectly happy using that for the tour I did last year (though I've lowered the gearing to give a 34x34 bottom gear). A race based CX, however might be rather uncomfortable over long distances.