It is a funding issue - vasectomy is more expensive then 15 years of mirena, and that's not allowing for the costs of treating those who fall into the 10% serious complications rate.
Many areas now ration this. Often by finding iOS to a certain level, but when that pot of money is spent, there will be no more in that financial year. It might be worth seeing if/when new funds will be released (April) and if you can be on the list for then.
But, as it takes around 4 months to get the proper all clear (and some men are much, much slower) you might not want to wait for that long. Lobby for more finding in your PCT for sterilisation is not likely to produce a quicker operation for you with, though might make it easier for those seeking procedures in future years.
Marie Stopes clinics have a good reputation in providing vasectomies, and doctors who do the procedure all the time (this reducing the risk of complication arising from error).
And also, as a minor issue, don't think of this as 'taking turns'. Turn taking is for reversible contraception, where yes, she's done a lot and your only choice is condones. But surgery is a whole different level, and neither of you has had a turn at that yet. You have however made a joint decision on what is best for you as a couple. That's an important difference, and in line with bodily autonomy (a very important principle)