I am not good at my indepth knowledge of the mid south of England but would suggest that before buying a property you actually rent somewhere to begin with. This way you will actually get to know the area better and have more time to find where is right for you. Rentals in UK should be readily available (except for the fact that Bournemouth area is, obviously a holiday area and some rentals may only be available for short term lets as these bring in more income). However I rally would suggest you go down this route. It is all too easy to think you know the area you want to be and then find that you have made a very expensive mistake as there is something wrong with the area that you didn't realise at first.
The basic time period for rentals in Uk is six months. At the moment in the British market a house purchase takes in the region of six weeks minimum to three months plus to complete. Estate agents (the name given to property sellers) might tell you different to this, but that is the reality. I know several people who have bought and sold properties recently and even when there is no chain and money is organised, nine weeks is the shortest period I have heard of in the last year. Others might say this is not true, but it can be a tricky business when the truth is not always told!
The New Forest area, near Bournemouth is particularly beautiful, and we did look to relocate there a few years back. Unsuccessful as we were unable to sell our current property at the time. Be aware that transport in this region is not brilliant! OK, you can get to London by road or rail, but not suitable for daily commute as very, very busy, and very, very expensive.

Good luck with your search. Don't dismiss renting. It is a common enough option and properties of all types and prices should be available. I hope you will be very happy. If you want any other advice on UK living, PM me.