A criminal record is for life. If any potential employer (for example) gets a CRB check on you, your conviction will always show up.
Criminal convictions do become 'spent' for most purposes after a certain period of time. For non-custodial sentences this is 5 years, not 4:
http://www.lawontheweb.co.uk/rehabact.htm
That means that, after 5 years, you can lawfully answer 'No' to questions on insurance applications (etc) which ask 'Have you ever been convicted of a criminal offence'. You'll also lawfully be able to answer 'No' to that question on most employment applications but not where that employment is exempted from the provisions of the Rehabilitation of Offenders Act (e.g. working with, or alongside, children or vulnerable adults). Also, as stated, some employers carry out CRB checks and your conviction would still show up, even if it was spent.
The Rehabilitation of Offenders Act doesn't apply to the requirements of overseas governments. The US authorities require visa applicants with criminal records to provide them with a copy of their police record. Convictions are never regarded as spent. With a recent conviction for a drugs-related offence, it's almost certain that you'd be denied a visa. In practice, tourists with criminal convictions can enter the USA, under the 'Visa Waiver Program', by simply keeping quiet about those convictions. (The US authorities have no direct access to UK criminal records). However, touring with a band requires a work visa, which you won't be able to get.
There's no form of appeal that will help you. Your criminal record is for life and it's extremely unlikely that the US authorities would consider granting you a work visa for far longer than the 5 year 'rehabilitation period' that applies, for most purposes, in this country. (It might be 10 or more years before the USA would