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Big Yellow Self Storage is a large storage company that has many locations around the UK. the nearest to coventry is in Birmingham. I've used them twice in London and my friends have as well (I used to study there) and they've been very helpful. Some locations offer extended access hours 06:00 - 23:00 and they should be happy to help you move stuff as well. I'd just give them a call and tell them about your predicament.
This is the only company I have used but as far as I know there are many many more. I think any large company will be very similar in service and I'd just give it a quick Google if you don't want to go with Big Yellow. Unfortunately that's the only one I've used so can't really report about any others.