@
casserole of nonsense
The national archives have a copying service but its absolutely extortionate, and i wouldn't recommend anyone using it, as they can often quote £200-£500 just for copying one file.
However, if you are short of time or can't get down to Kew yourself there's a few independant researchers i know who will 'copy' them (i.e. take photos of the pages within the file) for a fraction of the price (usually between £10 to £30 depending upon number of pages), and can provide a good service. If you want their details drop me a pm and i'll pass on their details.
The War Diary files that are likely to be of interest to you are as follows:-
WO 166/4758 INFANTRY: 9 York and Lancaster Regiment. 1940 May, 1941 Mar.- Dec.
WO 166/9029 9 York and Lancaster Regiment 1942 Jan.- May
WO 172/902 9 York and Lancaster Regiment 1942 June - Dec.
WO 172/2565 9 York and Lancaster Regiment 1943 Jan.- Dec.
WO 172/4937 9 York and Lancaster Regiment 1944 Jan. - Dec.
WO 172/7683 9 York and Lancaster Regiment 1945 Jan.-Nov.
WO 172/2561 2 West Yorkshire Regiment (Prince of Wales's Own) 1943 Apr.- Dec.
WO 172/4933 2 West Yorkshire Regiment (Prince of Wales's Own) 1944 Jan. - Dec.
WO 172/7679 2 West Yorkshire Regiment (Prince of Wales's Own) 1945 Jan.-Dec.
A lot will depend on when he switched units, but the 172 series will be the Burma ones and the 166 series will cover the time in the UK (and also Iceland).
You can also download a copy of his medal citation from the National Archives website for a cost of £3.50. There used to be a way to get hold of them for free, but not sure if they are still available that way, so would need to check.