Currently my IP space is announced by my old upstream and there ASN. We now have another upstream we are doing BGP into. Can I simply start announcing my IP space on this new circuit with my ASN then a bit later have the old upstream cease announcing my IP space with there ASN?

Will there be any harm having two different ASN's announcing the same space for a short period? It will certainly make the cut over easier. I can route all the incoming traffic from both into my network easily for the short transition and send all out bound too one or other.

That will not be a problem. BGP explicitly allows for this.

And it did work without any issues.