Hi Sophie,
Based on your requirement, you don't need to go to the expense of buying / leasing a phone system.
A 3 line Featureline Huntgroup will do exactly what you are looking for. Each line will have it's own geographic number, a 2 digit extension number (for transferring between extensions) and 1 number which will sit on top of all 3 lines (Hunt Group number).
When the Hunt Group number is called, your phones will ring in the order that you have requested - sequential hunting (the same phone will ring every time with calls being past to line 2 when line 1 is busy and so on) or cyclic hunting (exchange presents call to line 1 and then the next time a call is presented it goes to line 2 and so on).
Your only capital outlay would be the cost to purchase 3 x Featureline phones (not mandatory but recommended). These can be bought from the BT online shop for about £72+ VAT.
Hppe this gives you another, more wallet-sensitive option.