They say it's updated within 24h of publication. Which seems to be as up to date as you'll get.
There are http://services.parliament.uk/bills/ and http://bills.ais.co.uk/AC.asp for tracking bills before publication (the later one isn't working at the moment though).
So for example the 1st reading in the Lords of the Online Safety Bill was on 14th May and the text can be read at http://services.parliament.uk/bills/2012-13/onlinesafety.htm....