So to really fix this, we need to first address two of your claims:
1) You said, in effect, that measurementUpdated() would be called on the main thread (for you said both append and dropFirst would be called on main thread. You also said several times that measurementUpdated() would be called every 2ms. You do not want to be calling a method every 2ms on the main thread. You'll pile up quite a lot of them very quickly, and get many delays in their updating, as the main thread is going to have UI stuff to be doing, and that always eats up time.
So first rule: measurementUpdated() should always be called on another thread. Keep it the same thread, though.
Second rule: The entire code path from whatever collects the data to when measurementUpdated() is called must also be on a non-main thread. It can be on the thread that measurementUpdated(), but doesn't have to be.
Third rule: You do not need your UI graph to update every 2ms. The human eye cannot perceive UI change that's faster than about 150ms. Also, the device's main thread will get totally bogged down trying to re-render as frequently as every 2ms. I bet your graph UI can't even render a single pass at 2ms! So let's give your main thread a break, by only updating the graph every, say, 150ms. Measure the current time in MS and compare against the last time you updated the graph from this routine.
Fourth rule: don't change any array (or any object) in two different threads without doing a mutex lock, as they'll sometimes collide (one thread will be trying to do an operation on it while another is too). An excellent article that covers all the current swift ways of doing mutex locks is Matt Gallagher's Mutexes and closure capture in Swift. It's a great read, and has both simple and advanced solutions and their tradeoffs.
One other suggestion: You're allocating or reallocating a few arrays every 2ms. It's unnecessary, and adds undue stress on the memory pools under the hood, I'd think. I suggest not doing append and dropsFirst calls. Try rewriting such that you have a single array that holds 50,000 doubles, and never changes size. Simply change values in the array, and keep 2 indexes so that you always know where the "start" and the "end" of the data set is within the array. i.e. pretend the next array element after the last is the first array element (pretend the array loops around to the front). Then you're not churning memory at all, and it'll operate much quicker too. You can surely find Array extensions people have written to make this trivial to use. Every 150ms you can copy the data into a second pre-allocated array in the correct order for your graph UI to consume, or just pass the two indexes to your graph UI if you own your graph UI and can adjust it to accommodate.
I don't have time right now to write a code example that covers all of this (maybe someone else does), but I'll try to revisit this tomorrow. It'd actually be a lot better for you if you made a renewed stab at it yourself, and then ask us a new question (on a new StackOverflow) if you get stuck.