The woman that Walter Hartright marries is, to all intents and purposes, Anne Catherick, a spinster and therefore free to marry. She had committed no crime and therefore was not hiding from the law, having been placed in an asylum on a private basis. An 'enemy' could have objected on the grounds of her not being of sound mind, but they'd have had to find her first. Let's give Collins the literary licence for that not to happen.
Whichever name she was married under, by the time everything was 'found out', I doubt the authorities would have taken action against the pair of them, as she had been, in any case, a widow at the time of her wedding to Walter and hence free to marry.