Hi,
I think what you are looking for is not attainable (at least I can't see how) because the time change changes the filtration of the processes you are studying.
Nevertheless you can get by the transform below an equivalent process that the one you get by time change. (from X_t=\sigma W_t getting $Y_t=dW_t$)
Under some conditions on the volatility of the diffusion you can (by so-called Lamperti's Transformation) get a process with unit volatility + drift term, then use change of measure technique (i.e. Girsanov) to get back to pure diffusion term and you are done.
NB: You can adapt Lamperti's transform (+ Girsanov) to get a process equal in law to your time change in more general case.
with this operation you still don't get the same law under different measure (because you have to transform the process first) but at least you stay within the same filtation.
Regards,
Reference for Lamperti's Transform : Theorem 2 in
http://www.imm.dtu.dk/English/Research/Scientific_Computing/People.aspx?lg=showcommon&id=271164