Do you have alternative accommodation you can use between the selling of your current home and the purchase of the new property?
A lot of the timing will be down to the vendors of the new property. Do they have other offers? Are they willing to wait for some time for yours to close before really getting into the sale?
I'm not sure how common it is, but an agent told me recently that they would no longer be accepting solicitors' letters as proof of funds prior to accepting an offer. This was while viewing a house where a previously agreed sale fell through when it turned out the prospective purchaser was unable to obtain the necessary finance despite furnishing a solicitor's letter confirming they had the means to complete.
If you want a short window between closing the sale of your current home and the new purchase, you'll need to gain the trust of the vendor's agent, and put a lot of effort into ensuring your sale proceeds without issues and have everything lined up for the new purchase.