The £139 you have quoted is probably for an hour on their diagnostics which is normally £118. Your main stealer is clearly even more of a stealer if they are charging that.
Is your mileage flashing? What year is your car?
Blue and Me does have a habit of breaking. If your mileage is flashing then disconnecting the earth terminal of the battery for 10 minutes may fix it, if not then a proxy alignment is needed as the modules loose contact with the main body computer and the mileage flashes to let you know. A proxy alignment realigns and writes data to the BCM after reconnecting to each module, then the mileage stops flashing.
Neither of those are guaranteed to work as it may be your module is busted and no amount of £140 updates will fix it. These modules do not require updates however if you go to the Blue & Me website, you can download them onto a USB stick and do them yourself. The need for an update would not stop your blue and me or your head unit from working suddenly though.
I suspect if the radio is frozen, then the 10 minute battery disconnect may do the trick. Let us know how you get on.
Blue and Me units can be bought for less than £100 for a 500, not sure about the 500L as they are more likely different versions but have a look on ebay, you'll probably find one. It's not imperative that you get an identical one. As long as you get one from the same sort of age of car, it'll work. You will still need a proxy alignment as the mileage will flash once connected.
Do not go anywhere near a Fiat dealers for this.
