The Lincolnshires' 7th Battalion certainly became a Royal Artillery unit; becoming 102nd (Lincolnshire) Light AA Regiment Royal Artillery in 1941. It was very common for such regiments to retain their old cap-badges (or officers' collar-dogs) while simultaneously being referred to as RA.
I'm having a root through the other battalions - the 1st, 2nd & 4th Battalions certainly stayed as infantry.
I'm pretty certain that the shoulder-patches are for the Royal Artillery, although I cn't make out the lettering. The Royal Lincolnshires should have 'LINCOLNS' in white on red when serving as infantry, but this doesn't match:
The two-tone arm-of-service strip would also tend to indicate Royal Artillery (who had red/blue strips, while infantry had plain red):
The lad on the left was my Grandfather, so it means quite a lot to me finding out a little more about him and what his regiment was up to in WWII.
His full name was George Bernard Fryer - maybe this can help confirm that the lads in the picture are indeed from the Lincolnshire Regiment, if someone has access to roll calls or something and cross reference this name??
102nd LAA Regt RA served in NW Europe throughout the campaign as GHQ Troops.
1st, 2nd, 4th and 6th Lincolns served as infantry:-
1st in India and Burma (Arakan),
2nd in France (1940) and NW Europe,
4th in Norway, Iceland and NW Europe,
6th in France (1940), NW Africa and Italy.
8th Lincolns were converted to 100th ATk Regt RA then broken up with the batteries being converted to Mountain Artillery. Elements of 186 and 188 Btys were used to make up 474 Mountain Battery which served in 1st Mountain Regiment with 52nd (Lowland) Division in NW Europe