Even though Edward (David) was always pleading the poor mouth he left Wallis very wealthy and there would be no need for the Queen to pay her bills. They didn't even own a house, except for the Mill which was bought when they were both getting on in years. The beautiful house in Paris was given to them by the French at a peppercorn rent. Wallis stayed at the Palace for her husband's funeral but the whole royal family left to spend the weekend at Windsor although I believe they did ask her if she wanted to go too.
Stephen Birmingham in his book said "She (the Duchess) had become obsessive, almost hysterically so, about money, though it was hard to see quite why. Though royal wills are never made public, the Duke had died worth, by the most conservative estimates, well over $10,000,000, and Wallis, as his "universal legatee" had got it all, with no death duties to pay Shortly after the Duke's death, she sold three and and a half acres of land in Marbella for 82,000 British pounds, the Duke had sold his 4,000 acre ranch in Canada for 110,000 pounds and the Mill was sold by Wallis for nearly a $1,000,000." No there would certainly be no need for The Queen to pay any bills for Wallis.
Earlier on the same author mentioned the visit the Queen made to her uncle just before he died. "In the press, her visit was termed a "reconciliation", but it was really no more than a necessary duty call. Wallis had wanted to go to Spain, to Marbella, in order to be out of the country while the Queen was in Paris, But the state of the Duke's healthy had prevented their going anywhere."
I am sure the Queen could have shed a few tears for the Uncle David she had when she was a child, but for Wallis, no I can't believe this.