If you apply for a visa, the US Immigration and Nationality Act states that the US Embassy MUST refuse to issue you with a visa. However you can then ask that your application be referred to Washington as an application for a 'waiver of permanent ineligibility'. The MINIMUM processing time for such an application is 6 months but it can take much longer. (Someone posted here on AB to say that their partner had to wait 15 months before being told that his waiver application had been refused; he had two convictions for driving without insurance).
So if you intend to 'do things by the book', you'd need to apply now if you were hoping to travel at the end of 2016. (Remember that there's no point buying airline tickets until you know whether you've got a visa. So, unless you were prepared to pay full fare to purchase tickets at short notice, you'd really need to allow the better part of two years between the start of your application and your intended travel date).
While the US authorities have no direct access to UK criminal records (because EU law prohibits it) the British authorities still share a great deal of information under the guise of 'anti-terrorism' (even though the people whose information is shared are unlikely to be terrorist suspects). Further, any specific request for information from the US authorities for details of a visitor's criminal record will always be granted. Attempting to enter the USA illegally would, at the very least, see you imprisoned until such time as you could be put on a flight back to the UK but could also see you facing much more stringent penalties.
If you can afford to wait the better part of 2 years before (possibly) travelling, apply for a visa. (You'll have to get hold of a copy of your police record and attend attend an interview in London). Your chances of getting a visa might well depend upon the length of time that has passed since your convictions. (While the USA doesn't recognise the UK's Rehabilitation of Offenders Act, offences twenty years ago would probably be far less likely to matter than those from last year).
Otherwise you could 'chance it' and apply for an ESTA but risk the possible consequences of doing so.
Alternatively, just go to somewhere else. You can enter any of the other 27 Eu member states without having to worry about a visa.