You are most likely not going to get to the border, let alone being allowed to enter.
Before travelling to the USA you will need to apply for and be approved for either an ESTA or a US Visa.
During the application process for an ESTA, you will need to answer the following questions :
Have you ever violated any law related to possessing, using, or
distributing illegal drugs?
Obviously you will need to answer "yes" to that answer. Once you do, it is most likely that your ESTA application will be rejected. Once this occurs you can instead attempt to apply for a US visa, however this process as a similar question, and will likely lead to (at best) a long and drawn-out process before a visa is issued, if one is issued at all.
You may be tempted to answer "no" to this question, but keep in mind that the US and the UK do share criminal information, so in this case your ESTA application will still most likely be rejected, in which case you will be in an even worse situation due to having lied on the application.
Your only real option at this stage is to apply for an ESTA, answer truthfully, and hope that it is approved. If it is, you're good to travel to the US and will most likely be allowed enter. If it's rejected, you likely don't have sufficient time to consider any other options.
Note that although cannabis is legal in some US states, under federal law it is still illegal through the entire country.