Does a UK Citizen who overstayed on a tourist visa eligible for a green card?

No, you will not be able to do this within the US if you are over the age of 21. This is only available for immediate relatives of US citizens once you are unlawfully present in the US. Immediate relatives are parents of, spouses of, and children under 21 of US citizens.

Since you are no longer considered an immediate relative of your U.S. citizen parent, you cannot apply for adjustment of status unless you remain in valid immigration status. Instead, you will be required to return to the U.K. for your immigrant visa interview at the U.S. consulate, which will subject you to the 10-year bar because you have accumulated more than 1 year of unlawful presence in the U.S. However, you are eligible for adjustment of status if you have a U.S. citizen spouse or child over the age 20 who can petition for you.

As an unmarried son of a U.S. citizen, you can expect to receive an immigrant visa sometime in 2021. At that time, you will face the problem caused by your overstay in the U.S. since you remained in the U.S. more than 1 year after your visa admission expired, you are inadmissible to the U.S. for 10 years. It means that you cannot enter the U.S. on any visa and cannot adjust status even if an immigrant visa is available to you through a family-based petition. At this time, you can leave the U.S., wait 7 years for your turn to get an immigrant visa, then wait another 3 years until you become admissible or ask your mother to file an application for a waiver of inadmissibility (the 10-year bar can be lifted if it would cause your mother hardship that Attorney General would deem severe enough to warrant executive clemency) Under today's law, if you continue to remain in the U.S. illegally until your immigrant visa can be issued, it is possible to apply for a waiver without leaving the U.S., so long as the applicant has no criminal record, and the family petitioner can show sufficient hardship. However, this procedure is new, is established administratively (rather than legislatively),and might not survive until the time when you would need it. Employment-based green card application will not work for you because of the inadmissibility (there is no waiver of the penalty for unlawful presence in employment-based cases) Your only chance to receive a green card sooner is through a marriage to a U.S. citizen (it would also eliminate the inadmissibility problem.)

No. You may only adjust as an immediate relative, spouse of a U.S. citizen.