You just might be in luck. Although most (if not all) of the major roles include singing (and solos), there are several characters in Oliver! that are non-singing roles. Among them are
- Mr. Brownlow (aged 50-65), a long-lost relative of Oliver.
- Dr. Grimwig (aged 38-50)
- Charley Bates (aged 13-15), the Dodger's buddy and one of Fagin's gang of boys.
- Charlotte (aged 15-20), a funeral home worker.
Naturally, some directors will try to infuse all cast members into the chorus, so, though these characters have no solos, they still may be required to sing in the chorus. My guess is that if your acting skills are solid, a less-than-attractive voice will be a secondary consideration. I've been in quite a few musicals myself over the years, and Oliver! is a great story with some terrific songs. Break a leg!
I can't tell you a specific role because every production of any musical is different, due to the circumstances that affect that particular presentation. You need to review the script and casting requirements, then find out if the particular production you are auditioning for will be doing any adjusting of roles to meet unique demands. While most main roles in a musical, by definition, are going to involve being able to sing, there may be roles available for support roles that don't require singing. You may also be able to bring a less-musical kind of interpretation to one of the major characters, depending on the desires of the director. (Think of Rex Harrison as the Professor in My Fair Lady - he spoke in rhythm with the music rather than actually sang for most of his lyrics!)