That's a nice photo and probably a historical one, marking what is most probably a unique occasion.
It seems that there was a reception at the Palace for representatives of each of the infantry regiments about to be amalgamated under the 1957 reforms.
Pity the photo is indistinct but I guess that's the limitations of colour printing in magazines at that time. Apart from the SLI chap (obviously) the only regimental rep I can identify is the Royal Scots Fusilier (wearing Glengarry), back row left of the kilted soldier who is centre of pic.
I'd hazard a guess that the chap in the centre is Highland Light Infantry but there are two others in Highland bonnets who could be.
I think you're on to something. The label on the photograph is 'Merger 1959 [sic].' Thirty regiments were amalgamated down to 15, and there are 30 soldiers in the photo; 15 to the left and 15 to the right. I don't think it would have been too difficult to align the 15 on each side with their amalgamation counterpart.
In which case, you correctly identified the representatives from the Royal Scots Fusiliers and the Highland Light Infantry.
As you mentioned, it's too bad the photograph is so indistinct that individual regiments for the majority of the remainder cannot be identified.