Hi sariska, sorry, been away all weekend. Tried to reply on phone but obv didn't work!
Afraid I don't really know Croydon. I think the south of the borough is generally considered nicer, more leafy etc. You could try Purley, Sanderstead and even Warlingham itself. Sanderstead has good train links but don't know anything about schools in Croydon. TBH I'm not sure chi chi really exists outside Zone 3! Many areas further out are very family friendly and have good amenities, just not generally the same sort of posh delis, boutique shops etc on tap. Lots of countryside to explore though 
. Personally I think there's a an invisible line you cross somewhere around Penge / Beckenham in SE Lon where the 'London feel' disappears. On the flip side though you then start to get more value for money in terms of house/garden size for not much extra travel time. As you'll no doubt already know there's a similar change beyond Hayes in Bromley and in southern Croydon, where everything becomes more rural and villagey.
I think for your budget you should still have some choices but they'll involve compromise, so you'll need to decide what your real priorities are. As another poster mentioned, house prices have gone through the roof in London the last few years. Also, as you'd expect, prices reflect popular school catchment areas. Areas of SE London like Brockley, Forest Hill, Crystal Palace, Sydenham have all got very expensive since the new overland opened up. Even Streatham as someone else mentioned. All getting very nappy valley now - you'll probably find it quite different to 6 yrs ago. If cafe culture etc is very important to you then you should still be able to get a small house / decent maisonette somewhere more central, particularly if you're open to non-period and ex-council, but you're unlikely to also get a large garden / garage and popular school catchment. Have a look on Rightmove to get a feel for things. I guess the other thing to bear in mind is the commute to Warlingham - if you're more central then that could be more difficult too with the London traffic.
When we moved we decided a big garden was really important to us, plus 4 bedrooms, decent commute (near a station and within greater London) and good state secondary school were essentials. We just couldn't get all that for our budget where we were in central London, so chose to look a bit further afield and forego the other stuff. Every family is different though so have a good think about what's most important for yours.
BTW I see your kids are 4yrs old so assuming they're either already in reception or you'll be applying for a reception place after the January application deadline then you probably won't get much say in which school you're allocated unless you go private. Not much you can do about this, but it should also mean you have a bit more flexibility in terms of areas and therefore price 
.
Good luck with your move!