How practical would it be to field rewind a large power transformer that's suffered damage from a geomagnetic storm?
Is it practical?  Definitely.  It is done routinely for 100MVA-200MVA scale transformers, and I am not sure exactly what the largest rewind over done is, but I know 600MVA rewinding/repair is offered by companies in a few locations. 
As the transformers get larger, the time and cost advantages rapidly dwindle and become reasonable to a narrower and narrower range of circumstances.  There are other solutions as well, such as on-site removal of the entire winding assembly and replacement with a factory-fresh new winding, along with requisite on-site drying and testing.

I notice people in the comments seem to be under the impression that it couldn't possibly make economic sense to bring the equipment necessary on-site over simply shipping a transformer back to the factory, or waiting for a new one.  And they are correct in saying that for many kinds of transformer service at this scale, it does require bringing what is effectively a mobile version of the factory on-site.  However, that's perfectly ok.  In fact, bringing a very nearly the whole factory on-site is generally much more economical and timely than either waiting for a new transformer or 'sending it back' (this is almost never done) for service.  It wouldn't make sense to move the needed gear on-site to build one from scratch, but it does make sense to bring the gear needed to do significant service to large transformers on-site in many instances.

This is because of the massive transport costs of large substation-scale transformers.  They have a uniquely terrible combination of indivisible size and weight, (the 'indivisible' part being particularly important - transporting something huge and and massive is simple and straightforward as long as it divides nicely into pieces that a semi-truck can ship) and an almost comical level fragility for something that is a house-sized monolith of silicon steel and copper.  Everything inside is so heavy that it takes very little mechanical shock to ruin a winding clamp or winding or what have you.  Transformers are many things, and none of them involve being structurally robust.  If it isn't force in a known and designed direction from the intense magnetic fields that operate inside, it'll probably just break something in the transformer like tungsten carbide through wet tissue paper.  

The challenges involved are so severe that, at least in the US, transport alone accounts for on average about 20% of the unit cost.  Over the lifetime of one of these transformers, shipping one back to the factory for service would ultimately result in 60% of the transformer's cost being entirely transport.  

On the other hand, transformer factories are a piece of cake to move.  They can be disassembled and reassembled on site and the transport/replacement cost is such I high target that it is not that hard to beat.    

You have to keep in mind, however, that this sort of thing is about an extreme of an example of 'case by case' as one can get.  Numbers will need to be run, and on a case-by-case basis, it must be determined if there is any advantage (in cost as well as time) to performing a rewind (as well as other options, such as a replacement winding assembly + drying service) over waiting and paying for a new unit to be completed.  Often, the unfortunate reality turns out to be that the best and quickest option for many situations is to wait that year for a new unit.  And, fortunately, it also often is not the best, and months and a good deal of cost can be saved by performing on-site servicing.  

You have not provided nearly enough information for anyone to even guess at the feasibility in your specific case, and I for one am not even remotely qualified to attempt such a guess, nor should you attempt to use stack exchange as the vehicle for something like this. 

You will have to do some searching and see if there are any companies that operate in your country or area that offer rewind service for your transformer size.  Then, you will have to get in touch with them and and go from there.  

For example, here in the UK, which offers rewinding for transformers up to 135MVA.  

AGECO operates internationally, though it is not entirely clear what services they offer where, so again, you'll need to contact them to find out if it is economical for you situation, but they will do repair (including full re-wind)
on-site for transformers up to 600MVA.

Of course, it is not practical to do this without utilizing outside firms that are equipped and experienced in this.  

But, yes, it can be practical.  It isn't always, and it may or may not be practical for you, but you do have some options worth looking into.  And my list of a staggering 2 links is most definitely not an exhaustive list, its just the first page of a google search.  

What is a better strategy is to make use of transformer service companies which will do a great deal of preventative measures (like periodically removing any moisture that builds up, which is as much responsible for insulation failure as heat.  Heat is not so much of a problem if the insulation is kept very dry, heat + moisture will begin to break down the cellulose in the insulation, however).  There is also a great deal of informatics, monitoring, and general peace of mind services that can help to at least give you some warning of an impending failure so you can get a head start on that lead time for a replacement.  

And, it never hurts to repeat the old adage:

Hope for the best.  Plan for the worst.