Scatty, good luck with your search. There is an earlier thread that has similar queries to yours but I will address your named places directly also.
Datchet - lovely village, great value, HORRENDOUS aircraft noise (we stayed nearly three years and you really need triple glazing to make it bearable). Not much choice of service (lousy GP). Also it has an SL3 (Slough) postcode which is a real issue for the postcode-sensitive, ie, me. Very slow train link to Waterloo.
Egham - nice area, unappealing high street, also badly affected by aircraft noise, area is cut in half by train level crossing which will only get worse as times goes on. Train link also quite slow.
Wraysbury is quite small but some very nice properties, same issues with aircraft and train though I'm afraid.
Shepparton is more mixed, superb properties close to river, but quite a lot of scruffy places, can feel suburban, has a Staines postcode.
Twickenham is on the highway A316 which becomes the M3. It is split between two postcodes (TW1 & TW2) and also two areas (Twickenham and Whitton). Had undergone very rapid price rises. Good schools, great local borough (Richmond), lovely area esp near the Thames, but cramped and overheated right now. Whatever is in your budget range now will be well overbudget in 6 months time. Think £500K for a 3x1 or £650K for a 4x2 semi in the least good part. We lived in Twickenham for three years but got tired of onstreet parking and living in a semi, so moved out of the postcode to a detached house.
Look hard in Hampton itself (TW12) there is more choice and a bit more space than Twickenham, but the market is still quite strong. If you're prepared to drive to Richmond station, there is tube and overground links, very handy indeed. Richmond borough residents get free parking if they have an economical car, a wonderful advantage in the borough. For getting to the school and Liverpool St, this area is the best logistically, although none are great.
Epsom is no where near all your other areas but driving to Hampton from there every day would be a good hour, a horrible drive, and not recommended!
You don't mention Laleham (part of TW18) which is lovely, the same cute village feel as Datchet without the godawful plane noise. Would recommend a look there.
Also Staines, it's suburban but friends with families are very happy there and it's good value. Train link still slow, about 45 mins I think?
Finally Lower Sunbury (TW16) is worth a look. Lovely near the river. Avoid the part of Sunbury north of A308 and the devastatingly ugly Sunbury Cross, but look close to Thames St. Very slow branch line (I think, I don't use trains from Sunbury as the road links are great) though and not in Richmond Borough.
Good luck in your search, it's very difficult from abroad (we did that too - I researched for 18 months before moving!). I have a lived and worked in these areas and wrestled with each move like mad before taking the plunge.