Below is some code to show how it might be done, hopefully there are sufficient comments.
As far as I know, some parts of the UK observe daylight saving but the following should not be affected by any change of timezone since it uses local date methods.
Any date can be used for the start of the cycle, I've chosen 10 June 2011 totally randomly. Also, any time of day can be used to change the message on the final day from "put your bins out today" to the standard "next cycle is in x days".
The algorithm is calculate the number of milliseconds to the next full fortnight from the start date by subtracting the number of milliseconds from the last full fortnight from the milliseconds per fortnight. Then the remainder is converted to full days.
To save ambiguity on the day before the cycle ends (in this case, a Thursday) the message says the cycle ends "tomorrow" rather than in 1 day, and on the day itself, up to 18:00 or 6 pm it says the cycle ends today. After that, it says the cycle ends in 14 days or less.
// Date objects are based on milliseconds (8.64e7 per day)
// Calculate milliseconds for a fortnight
var msPerFortnight = 8.64e7 * 14;
// Any date can be the start of the fortnightly cycle
// Make Friday 10 June 2011 first day of cycle,
// Cycles end on 25 March, 6 April 2016.
// Note months are zero based so June is 5
var firstDayOfCycle = new Date(2011, 5, 10);
// Time of day to stop showing "put bins out today" message on start/end day
// 18 is 6 pm
var endHour = 18;
// Get the day name for the cycle start/end
var dayName = ['Sunday','Monday','Tuesday','Wednesday','Thursday',
'Friday','Saturday'][firstDayOfCycle.getDay()];
// Convert end hour to convenient format with am/pm
var endHourNeat = (endHour % 12 || 12) + ' ' + (endHour < 12? 'am' : 'pm');
// Get a date for now
var now = new Date();
// Get milliseconds to next full fortnight by
// msPerFortnight minus milliseconds since last full fortnight
var m = msPerFortnight - ((new Date() - firstDayOfCycle) % msPerFortnight);
// Calculate time remaining full days
var daysLeft = Math.ceil(m / 8.64e7);
// Create a human friendly message
var message;
// If more then one day left, or after 18:00 on last day, show this message
if (daysLeft == 14 && now.getHours() < endHour) {
message = 'Today is ' + dayName + ', so please put your bins out before ' + endHourNeat + '!';
} else if (daysLeft > 1 ) {
message = 'Put your bins out on ' + dayName + ' in ' + daysLeft +
' day' + (daysLeft == 1?'':'s') + ' time.';
} else if (daysLeft == 1) {
message = 'Put your bins out tomorrow, on ' + dayName + '.';
}
document.write(message);
You can even add the date of the end of the cycle and add ordinal, so it might read "Put your bins out on Friday the 25th, in 3 days time". But I'll leave that up to you.