I will be travelling to the United States on 15th September. First port of call is New York City... then on the 19th we wil be flying to Vegas for the weekend before returning to NYC on 22nd.
We return to London on the 25th September.
Will I need to apply for a visa anytime during this trip? Thanks.
.. and most important of the 'conditions' relates to convictions in the UK (or elesewhere). Even if you have convictions in the UK the US authorities are unlikely to have a record of them. So since you have no prospect of getting a visa now (if you now thought you needed one) you'll just have to fill in the I-94 visa waiver form on entry and see what happens.