Hi Groovy yank
That is a very large increase and I'm not surprised at you feeling shocked. What strikes me as a first impression is that the landlord is being greedy in order to cover his/her mortgage payments - and thats probably due to the fact with all the uncertainty in the mortgage industry, his/her 'discount rate of x years' is coming ot an end soon and any new mortgage payment they have to make is substantially higher.
The FIRST thing you need to do is investigate and research thoroughly what similar apartments/flats in your block, street and area are going for - the closer to you the better. This will give you a good basis in writing back to the landlord to show what the going rate is. If it's roughly what you pay now, then a £125 increase is way too greedy and you 'should' look for other places to go (i know thats easier said than done - but why should you be forced to pay over the odds in the same area?) if the other places are indeed 90, 100, 110 per month more than you, then be prepared to pay more to stay in the place.
Personally, my rental apartment achieves the absolute maximum in the area it's in, there are exactly the same flats as mine which get £50-75 less than mine (or at least when i last talked to the other landlords) as I totally refurbished it, put in high quality items, total new bathroom and kitchen and many many other aspects too. If I ask for more, then it makes sense for the tenant to move to another part of Slough where the area is better and they can rent one of these fancy new build flats for only 50-75-100 per month more.
If you can assess your place and point out any flaws if need be that could go to your negotiations with the landlord. I obviously don't knwo the reasons for such a large increase in rent, but I strongly suspect it's becuase they have to remortgage and the rate their getting isnt so discounted - putting them in more of a 'reality' situation.
Stand your ground, and if they're adamant you pay it even if it's way beyond similar place rents then really do consider moving out - but make sure you point out to them it's they who'll have an empty unrented place and if you can negotiate a better rate it works out better for them to keep you there so they dont fall back on mortgage payments!
I don't know too much about the box storage places and moving costs etc, but for just a mile away, it may make more sense to find a man with a van or hire one for the weekend yourselfs and make lots of to and fro trips..
Good luck!
DtM! West London & Slough UK!