Yup not a chance of a royal match for May as she was known in the family.
She was popular with the Queen who admired her given the rather difficult life she had with her spendthrift and loud parents (the Queen was always a bit jealous of Mary Adelaide's popularity with the public).
Victoria herself had asked about the possiblity of May's father succeeding to the Wurttemburg throne (given the succession crisis there) largely to help the Teck's out of the financial mess they were in but it was pointedly rebuffed and she was reminded of his morganatic status.
George V's personal desire at the time had led him to another cousin -Princess Marie (Missy) daughter of his uncle Alfred - Alfred and the Prince of Wales were quite keen on the idea though she was a good bit younger. Missy's mother Marie of Russia (who had found the British Court so difficult) was very quick to find an alternative and the teenaged Missy was soon married to the King of Roumania's nephew.
George's brother Albert Victor had briefly thought of marrying Alix of Hesse though and the Queen was keen but Alix refused.
One of the things that is so often forgotten is that George V did not automatically inherit May.
After his brother's death, he actually proposed to Missy and upon the instruction of her mother, she declined. She was quickly rushed off to Berlin and forced to accept a proposal from Ferdinand of Romania, who in the Duchess's eyes was a more acceptable candidate, simply because he wasn't British.
The weird thing is the Duchess genuinely liked George and the feeling was reciprocated, when she died in 1920, George wrote that Maria Alexandrovna had been like a second mother to him, partly raising him, while he lived in Malta where the Edinburgh's were based in the 1880s.
Missy's refusal caused a very big stink and a falling out between the Prince of Wales and Duke of Edinburgh, even though in fairness the Duke would have loved to see his daughter as the future British Queen.
Victoria only pushed George towards May after the whole debacle to try and move matters along and get George married.