No way. There was a huge scandal after King Henry died, that Elizabeth was pregnant by her stepfather, Thomas Seymour, the new husband of Henry’s last wife Catherine Parr.
Also, there was no privacy in those times especially for a royal. There were spies everywhere and there is no way that this secret could have been kept into adulthood. When she was stricken with smallpox, it surely would have been discovered. Since there was such a drama about her successor, had this been the case, Catherine Grey would have immediately been crowned as the new ruler of England, especially with the power of the Seymours behind her.
You're absolutely right, and I highly doubt that chief spy master Francis Walsingham would have looked the other way on this.