There is a classic book: “Physical Chemistry of Macromolecules” 1961 by Charles Tanford, John Wiley & Sons Inc. The book has a particular section that discuss the relationship between the diffusion coefficient and the molecular weight of the polymer. If you know the molecular weight of the monomer, you could estimate the molecular weight of your polymer from values of the diffusion coefficient. The author gives some interesting examples of D vs M dependence and useful references.    

Can you conduct a GPC ?

When talking about "range" do you mean estimation of molecular weight DISTRIBUTION or only the AVERAGE? Distribution requires more data. Additionally each average estimation method yields another kind of average (e.g. number average,  weight average viscosity average, z-average) which are not necessarily comparable with each other ...

The best thing you can do , in my opinion , in order to determine the molecular weight of your polymer is GPC. There are some GPC which can also determine the absolute molar mass.What I would do is to inject both the monomer and the relative polymer to have an idea about the Mn and the polymer dispersity compared to the monomer unit. It is also important to know the GPC system you are using and the compatibility with your polymer. (the standard used, the cut off of the columns..).unless your GPC system is able to determine the absolute molar mass of your polymer, what you'll know is anyway a molar mass due to the hydrodynamic volume of your polymer, ones injected.

I am only familiar with rheological and SEC/GPC  methods to measure molecular weight distribution, but 3 years ago I came across an information by Jean-Philippe Starck (CEO at NMRTECH ) about your question:

"NMR and poyldispersity index of polymers (PDI)

The DOSY method measures the translational self-diffusion of molecules in solution and allows a precise analysis of a pure product or a complex mixture without any prior preparation or separation.

Compared to other PDI determination technique such as Mass Spectrometry or Size Exclusion Chromatography, this approach presents the unique advantage of a direct measure which does not require any interaction with a static phase, separation, ionization or dilution of the polymer. It does not require any special calibration, equipment, or preparation and is a rapidly obtained with a diluted polymer sample.

For more information: J. Magnetic Resonance 212 (2011) 169-173 or contact me."

Available on researchgate."Polydispersity index of polymers revealed by DOSY NMR"

I would be interested if anyone could share his experience with this method.

Depending on whether you are measuring the chain diffusivity in a melt or in (dilute) solution, you might employ some scaling relations to estimate the molecular weight of your polymer. However, you can only do this relative to a sample of the same polymer where both molecular weight and diffusivity are known.