The way the law works in these cases is that regretting committing a crime is not relevant when a court decides whether someone is guilty or not. It might reduce the punishment imposed, but not the fact of guilt, and the conviction that results. If TfL prosecute, which is very likely, the usual advice would be to plead guilty because of the lower fine which will be imposed.
The interaction between DBS checks and criminal convictions is complicated and difficult to explain, particularly where very minor crimes are concerned. In many cases this sort of conviction won't appear on a basic DBS check and might not appear on others. But employers might ask - and have the right to ask - questions about convictions, rather than about the records of convictions disclosed in a DBS check, which makes things very complicated, and often leads to the advice to be open and honest with employers, rather than relying on what is or isn't reported by DBS.
It may be wise for you to consult with (and join) a union, or to contact charities which work with people with criminal records (such as
https://unlock.org.uk/) or to get advice from people who know more about the regulation of care workers and the attitude which relevant employers have to minor criminal convictions. This might help you to understand what the best approach is to take with your current and any future employers.