I can't address legality but I don't think it's good practice for you, as an individual, acting independently of your employer, to notify your employer's clients. It's really up to the employer to decide the method, content, and timing of that message. The relationship is theirs, not yours; especially now that you're leaving. They have to maintain the relationship once you're gone, not you. The transition is their business, not yours. They very were may want the email to come from you, but it shouldn't have come from you without their consent, and without including some sort of reference to the transition (ie "so-and-so is copied on this message and will be handling your relationship).
Even if you did not explicitly give them personal contact info or solicit their business at your new employer, it's pretty easy to see "hey I'm leaving!" and take it as a reason for them to follow up with you with respect to your new employer - especially since there wasn't an explicit hand off to someone else at your old employer to handle the account. You've left it totally open ended, which is going to be problematic for your old employer. Their clients are now in doubt and don't have a clear plan for going forwards.
All that said, I've had situations where I developed close, personal friendships with clients, and I spoke with them about changes in employment directly. However, that was done through "friendship" channels (meeting them for dinner, a personal call to a personal phone number after the work day was over, etc), not through professional channels (email from my work account, phone call during the work day from my work phone). If you don't know these clients well enough to be contacting them via personal channels after hours, you don't know them well enough to inform them of your departure without your employer's consent.