It doesn't look as if either if their service records have survived. As you may be aware approximately two thirds of all the records of First World War soldiers were destroyed in a fire during WW2 following a bombing raid. However the records of the medals they received should have survived but I haven't found them so far. Assuming they both went overseas they should have got the British War Medal and Victory Medal (Squeak and Wilfred, not Alfred). I am surprised that Frederick Mark Taylor was promoted to Colour Sergeant - the rank of someone holding the appointment of Company Quarter Master Sergeant - in just three years. It's not impossible for him to have done so, however I would suspect he may have had some prior Army experience, maybe in the Territorial Force or Militia before the war started, for such rapid promotion.
According to the
Long, Long Trail website, which can usually be relied on as very accurate, none of the Leicestershire Regiment's battalions were sent to Greece during WW1. The only one which went anywhere other than France or the UK was the 2nd battalion which spent the latter part of the war in Mesopotamia and Palestine. It is of course possible that although he enlisted into the Leicesters, Frederick Mark Taylor was transferred to another regiment. As far as I am aware the only British troops who took part in the Thessaloniki Expedition were the
10th (Irish) Division, whose infantry were mainly Irish or Canadian Regiments, with the only English battalion being the 10th Battalion of the Hampshire Regiment.
There is a militia attestation in the Leicester Regiment dated 13 Nov 1880 for a Frederick Mark Taylor of 42 Dorset Street Leicester, a shoe finisher aged 17 years and 9 months. Could this have been his father?
The only Frederick Newick entries I found referred to a Gunner Newick of 301 Siege Battery Royal Garrison Artillery who received the British War Medal and Victory Medal.