Your husband won't be able to enter the USA legally. The US embassy website states that anyone who has ever been arrested (even if totally innocent) or who has been convicted of any offence (except very minor motoring offences) is ineligible to enter the USA under the Visa Waiver Program. (Your local police officers seem to be referring to the provisions of the Rehabilitation of Offender Act, which don't apply under US law).
Your husband is (officially) obliged to apply for a visa. This involves getting hold of a copy of his police record (maximum waiting time 40 days), obtaining an appointment at the US embassy in London (typical waiting time around 2 to 3 weeks) and then waiting to hear the result of the application. (One page of the US embassy website quotes a minimum of 8 weeks. On a different page, the minimum waiting period is given as 14 to 16 weeks). So it's obvious that your husband has no chance of obtaining a visa before the beginning of your employment period. It's also very unlikely that he'd get one before you finished your period in the USA.
If your husband wishes to visit you, he'll have to lie on the visa waiver form (which travellers complete on the plane). US immigration officials don't have direct access to UK criminal records. Although some information is passed on, it's unlikely to include relatively minor theft offences. So he stands a very good chance of being admitted.
Chris