Hi Zoe
Welcome to the forum.
The Grenadiers and Coldstream both had shoulder titles with white lettering on a red background so it's not clear from the photo which he might have been. The only difference is that the Coldstream used sanserif lettering, whereas the Grenadiers used letters with serifs. Unfortunately the photo isn't clear enough to show that sort of detail. If you have the original photo rather than the scan, try checking it with a magnifying glass.
I'm not sure what your dad meant about the buttons as there aren't any that I can see in this photo. I assume he was talking about the spacing of the No1 Tunic (the scarlett ones the Guards wear on ceremonial duties). If so then that doesn't really help here since your grandad is in battle dress.
Almost uniquely the Scots Guards and the Coldstream retained their own records for the period after 1920 so if your grandad was Coldstream you may be in luck tracking down his details. You need to contact their
Regimental Headquarters with as much detail as you can supply. Sadly the same does not apply to the Grenadier Guards and the only place you will find their records is with the MOD. As you say it's difficult to apply for any service records without a death certificate but presumably you can get that fairly easily.
If you haven't already done so, I suggest you try putting his name into a search on the National Archives website along with the word Guards and limit the date range to 1939 - 1946. Something may just crop up, such as a medal award or casualty list entry.
You could also try searching for your grandad's name on the WW2Talk forum. They are assembling a list of known regimental numbers for the two regiments and you might just get lucky providing that his was a not too common surname.
http://WW2talk.com/index.php?threads/coldstream-guards-reference-thread.80210/http://ww2talk.com/index.php?threads/grenadier-guards-reference-thread.80209/If you post his name here, I'm sure some of our talented sleuths will find some more details for you.