Laser surgery will only correct one, realistically, your normal laser surgery, same on both eyes, will correct normal vision problems but not the old age reading issues.
But they are starting to do different things now like you have both eyes adjusted for different depths of vision, so one is set a little short sighted to combat the presbyopic issue and let you read with one eye taking on the burder, while the other eye is set to normal or slightly longer distance so the other eye can see clearly for things like driving and anything else at distance. Should read up on people who have had it. I know some of them will use a pair of glassed with opposite type lenses so when you want perfect vision at normal distance for say, watching a film, you can slip on glasses to bring the nearer vision eye to medium distance, and the long distance eye also to medium distance.
Its really an older person issue, the massive majority of normal long/short sighted people will have similar vision problems in both eyes and will be corrected towards the "normal" with laser eye surgery. THe problem is presbypoic's is an age thing and reduces reading quality even with the rest of vision being normal(or normal ish).
Xstyle, if I were you and wanted a second opinion I'd go to moorfields without question, if you can get an appointment before you need to confirm, or before you can cancel the data for the other Op i'd certainly at least see what they have to say.
Moorfields I think personally are the choice for anyone whose slightly out of the normal boundries of treatment, so those with higher perscriptions, or thinner cornea's, or any other slightly not normal issue. I'd also take Moorfields word as almost final, or at least if 2 docs from Moorfields agree'd the corner is simply too thin to do it safely I wouldn't do it, full stop. But thin doesn't mean impossible and just because anyones said thin cornea I don't mean its hugely risky.
Just with all the stories about cowboy branches out there that will operate when they really shouldn't, Its a good safety check I guess if their opinion matches up with a Moorfields guy. If it doesn't I'd stay well clear, if it does, then the choice between prices and who you feel most comftable is up to you.
I'm fairly sure most people get retreatments at Moorfields, and most other places also, as long as the problem is fairly obviously from the surgery or the eyesight simply hasn't improved enough, a repeat surgery won't be done that quickly as the eye needs time to fully heal as you can have improvements, albeit minor, over 4-5 months even if the majority of change is often in the first few seconds. Generally if they miss the target vision quality, or introduce massive halo'ing/night vision problems you'll know fairly quickly, it would be monitored and they'll attempt to fix it without any additional cost.
AFAIK its incredibly rare for you to have good or great vision, and 2 years later suddenly have issues, I've not seen anyone complain about that kind of thing yet.
Depending on your age it might not be worth it though, because most of us will have vision problems as you age ,normally mid/late 40's to late 50's when it starts. So if you're 40 or so you might find the surgery works, but you get other problems very shortly after. At 27, I'm going to have it done sometime in the next 6 months or so, just waiting till I have enough to pay for it basically now. I should get anything from 15-25 years before I could need glasses for presbyopic issues which is good enough for me.