For telephone claims, your date of claim is the date of the initial telephone call to the claim-line, as long as you provide the required information during that call or within a month of the call (or longer if the decision maker considers it reasonable).
UCPIP(C&P) Regs , reg 14

Advance claims – If you are receiving statutory sick pay (SSP), you cannot claim ESA at the same time. However, if you know your SSP will be running out, you can make an advance claim for ESA up to three months before your SSP expires. This can help to ensure minimal delays in receiving any ESA to which you may be entitled.
UCPIP(C&P) Regs, reg 34

Late claims – Your award of ESA can be backdated for up to three months before the date of your claim if you claim for that earlier period and would have met all the qualifying conditions during it. You will need to ask your doctor, or other healthcare professional, to confirm on the fit note that you were not fit for work during that earlier period