[QUOTE=Hypno-Toad]
Parliament is adjacent to Westminster Abbey and both are about a half mile south of Trafalgar Square and the National Gallery. Downing Street is between the two. So is Churchill’s wartime bunker, a great little tour. THAT is on the eastern edge of St James Park, which has Buckingham Palace on the western edge and the Wellington Barracks with attendent Guards Museum on the southern side. I think that this paragraph all by itself would constitute a full day and then some.
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I was going to make this point - if you wanted to be efficient you could do the following in one day:
Trafalgar Square (half an hour tops is all you need once you’ve taken some pictures)
National Gallery (as others have said you’d need a lot of time to enjoy this properly but it’s just next to Trafalgar Square)
Hyde Park, and possibly St. James Park (St James I heartily recommend - it’s just next to where I work - as it’s pretty and has nice things in it but ultimately it’s fairly small and when all is said and done it’s a park. Hyde is nice but it’s very large and not much to it, so you could skip it)
Buckingham Palace (this is in St James Park - do you want to go inside and take the tour which runs seasonally? If not half an hour tops, if so as long as the queues take and probably another couple of hours)
Big Ben/House of Parliament - really worth having seen but unless you’re planning on going inside (which you can do, quite a lot of the main House is a public area and on Fridays they do a tour which I’ve been on and recommend - you could even take in one of the Parliamentary committees whilst you’re there!) all you need is time to take pics - half an hour at most
Westminster Abbey - next to Parliament, again if you’re not going inside you don’t need long to take pics
The London Eye (maybe) - I’ve been on it and it’s good (and not too expensive) and it’s only across the bridge at Waterloo from these other attractions. The actual ride is half an hour and you can buy a ticket for a set time rather than having to stand in line which is a bonus.
All the other things you list are in different parts of the city so would take longer to do and of them I’d say Natural History Museum was the best, but that’s just me. If you asked me to break your trip down as you have it I’d do it as follows:
Day 1
Trafalgar Square (take pics) then go in National Gallery for 2-3 hours followed by lunch then head to Covent Garden piazza for a quick mosey, and then down to the Embankment and walk across the bridge to the London Eye for the afternoon
Day 2
Go to Westminster tube stop and take some pics of Big Ben and the Houses of Parliament (see how long the queue is to go in and if it’s short go through, check out Westminster Hall and Central Lobby, have a quick wander around the corridors if you like (police will stop you going anywhere you’re not supposed to) and then exit pronto - at the very most an hour), then cross the road and do the same for Westminster Abbey but don’t go in as it costs money and isn’t really worth it. If you wanted to you could walk around the corner into Great Smith street and come visit me in my office as I’m just around the corner from the Abbey. 
Cross the road and go into St. James Park, do a circuit of it circling from the bottom right hand corner up past the back end of Downing Street and Horse Guards, along the Mall to Buck Palace, take some pics, maybe wait for the guard to change if you’re really keen to see that. Now, lunch time! Go out of St James park on the North Side and up St James’ street to a restaurant half way up on the right hand side called Just St James. It’s not a cheap place to eat (lunch would be about £25 per person for two courses) but the food there is amazing, the venue fantastic and the service second to none - it’s my fave London restaurant and it will leave you with a great impression of London dining.
After lunch walk the rest of the way up St James Street to the Junction at the top and turn right and walk along until you get to the Royal Academy of Art on the left hand side of the road. Pop in and if there are any exhibits that take your fancy then check them out (they’re usually not that pricey and you can do them in about an hour) but if not then exit and carry on along Picadilly to Picadilly Circus and Leicester Square (yes they’re tourist traps but they’re worth having seen). Don’t buy anything there unless you want to be ripped off. Whilst you’re in that area check out the multifarious theatre outlets and see if there are any tickets for shows you’d like to see that night that aren’t a complete gouge (not likely but you never know).
Day 3
In the morning go over to the Tate Modern and have a look around, 2-3 hours is the most you really need. After that walk over the bridge to the City and pretty much a straight line north to St Pauls. Go in and have a look around for an hour, but I don’t recommend going into the belfries if you have even a mild fear of heights. For lunch probably just best to grab a sandwich from a Pret or Eat! or something.
This next bit might sound very strange but bear with me: you’ll now be in the City of London which has some of the most amazing arcitecture in the world and is effectively a maze of streets some of which are nearly 1000 years old. Put the map away and just walk around and take in the sights, it’s really worth it. Once you get tired of doing this stop at the next tube station you find (there are a lot in that area) and take the tube to Tower Hill where the Tower of London is. Go in and have a look around, as cormac262 said you can do this in 3 hours at the most. It’s one of the most historic sites in London.
Day 4
Head to Natural History Museum at South Kensington in the morning. I recommend 3 hours there and then that leaves you the afternoon to check out the British Museum.
Job done. 
