Adrian,
The cap badge is definitely that of the Royal Artillery (RA) but unfortunately that is one of the largest regiments in the British Army. If your father joined the army as a member of the RA before 1943 his army number would have fallen in the sequence 721,000 to 1,842,000 as that was the block given to the RA.
The arc at the top of his arm is a what is known as a shoulder title for the "Royal Artillery" which would have been those words in red on blue. Below that he has the formation sign of 148th Independent Infantry Brigade which is the letters NM joined. This was originally a brigade of Territorial units from the North Midlands hence NM as their badge.
In July 1942 the Brigade was withdrawn from the active army to become a training formation. Its job was to train potential officer cadets before they were sent to Officer Cadet Training Units or OCTUs. It was located in South East England.
By 1944 the NM sign would only have been worn by those doing the training (not the potential officers) so it seems likely that your father was one of those involved with the training. He has no visible rank badges so was either the RA equivalent of a Private - known as a Gunner - or he has a rank badge on his cuff which we can't see.
Because the Brigade was a training unit the units in it would have been a strange mixture so its is likely to be quite hard to be exact as to which regiment of the RA he was serving with. Without that it's not going to be possible to take this further
Each formation of the Army was required to keep a diary known as a War Diary, but none appears for 148 Brigade on the National Archives website. There is one file which covers the conversion of the Brigade to a training role which may help. Its reference is WO199/585. I can recommend researchers if you'd like it looked at but of course they charge.
Have you tried to get his service papers as next-of-kin?
Hope all that helps. Shout if anything is not clear.
Any chance of a 300 dpi jpeg for my files please?
Jon
Postwarden