Hiya
I have no experience of the lastminute.com website to offer I'm afraid, but as frequent vistors to London ie 10 times plus per year we have always, without fail, picked up tickets for same day Westend show from either the actual theatre Box Office, they are all pity much next to each other, or from one of the various Ticket Offices in and around Leicester square etc.
Only very recently we travelled down to London on a Saturday and ended up staying over. We managed to get 4th row tickets to see Dirty Dancing from the Theatre Box Office.
London is fantastic, enjoy