Yes no building regulations for floor tiles. But there are NHBC warranty standards.
Workmanship and quality are, unfortunately, subjective. But the NHBC tolerances do put a figure on what is deemed unacceptable.
They do make reference to the NHBC warranty but only in so much as to the registration of the house with NHBC and that the buyer must pursue appropriate claims for defects with the NHBC and not the seller.
You may have a case against your solicitor for not highlighting this. It is a big red flag warning that Elan homes appear to be not interested in dealing with defects in the poor quality homes they build and sell.
Regarding the SAR and I'm not an expert but have applied for SARs myself, there are very few reasons that an organisation can legally withhold information with you name on. So again Elan are being difficult for no reason as the ICO will after around a year, compel them to comply in every respect.
You can always write the the NHBC and request a SAR from them. Might be interesting.
What Elan Homes are prepared to give you is sales and handover paperwork. You have a legal right for copies of all Building Regulation inspections carried out during the construction of your new home, whether your name is on them or not. These may have been done by the NHBC.
If nothing else, I do hope your post and experience acts as a big red flag warning for others even considering buying a new home from Elan Homes.