Don't use the Tube - not because you're scared of it, but because you get to see so much more of London above ground by walking and on buses than underground and that's the point of a trip.
The tube is really handy for getting around and isn't dangerous or uncomfortable off-peak, but you miss so much and that is what you've come to London to see. That's a tragedy. So use it strategically.
I agree with all the people who say get an overground train from Croydon into Victoria. It will take 30 mins and then you can easily walk to loads of tourist places.
Buckingham Palace and Green Park and St James Park are under 10 mins walk from the station. You can walk through either park, which will be lovely if you get a dry autumn day, and end up in Piccadilly in 10 minutes' more and from there you can explore Leicester Square, Covent Garden and Chinatown to the east. People sneer at Leicester Square but it's been cleaned up and kids like it.
Westminster Abbey and Parliament are about 15 mins from Victoria in a different direction, although it all connects up - have a look at a map.
From there you could go north on Whitehall past the Cenotaph into Trafalgar Square to Covent Garden...
But it's 5 minutes' walk south from Parliament Square across Westminster Bridge to the London Eye and then you can continue onto the South Bank and then you can walk east all the way down to Tower Bridge and there is loads of stuff to see and do that would appeal to children and adults.
I wouldn't recommend walking all the way down to Tower Bridge from Victoria - it would take about two hours - and that's just walking reasonably slowly but not stopping.
But if you walked to the South Bank for example, you could cross back to the northern embankment by Waterloo Bridge, said by many people to be the loveliest view of the Thames, to Embankment tube which is a completely bearable 4 mins to Victoria and then home again in time for tea in Croydon.
Someone mentioned the 52 bus from Victoria Station. It's brilliant. I'm sure she's right. It probably does stop near the Natural History Museum but I'm going through the route in my head and can't think. I think she must know where to get off. Ask her.
The C1, which is a little bus, goes from round the corner from Victoria Station to the NHM and beyond, if you're worried about getting lost.
But the 52 route is much better than that. Sit on the top deck at the front, right hand side if you can.
You'll go up to Hyde Park Corner and see Wellington Arch, the Royal Artillery Memorial and No 1 London, which was the Duke of Wellington's house. You'll turn at the Lanesborough (look out the left) which used to be an NHS hospital, but what the hell, London needs more luxury hotels 
, then it's the southern bit of Hyde Park, Harrods (left hand side), Albert Memorial and Royal Albert Hall (look out the left), Kensington Gardens, Kensington Palace, then it goes off to Notting Hill and Portobello Road if you want. It is a free sight seeing tour.
If you take that route any time between 9.30am going out and 3pm coming back it will probably take 30-40 mins to get between Victoria and the NHM. People go on about the traffic but in those times you'd be unlucky to hit bad congestion. I know it would be quicker on the tube but you'd miss so much.
If you don't want to waste time on the bus back just come back from South Kensington tube which is a minute from the NHM and only about 6 mins to Victoria.
Have a lovely time.
PS You are not a wuss for being daunted by the Tube. I'm not daunted but I'm glad I don't travel to work on peak because people really get on my tits .
PPS sorry/not sorry for the length