The A38 one is good enough that I, as a confident cyclist who isn't afraid to ride on the Bristol Road, have been using it to get to and from the centre of town from the bottom of Selly Oak since it opened. It's slightly slower at quiet times, but quicker in rush hour because you don't get held up by traffic. I've seen an average of about 10 other cyclists on it each time I use it (varies greatly with time of day, and the students haven't come back yet), about half of which are of demographics I wouldn't normally expect to see riding (even pavement cycling) on that route.
Here's a video:
https://www.dailymotion.com/video/x7bq5llThere's a similar lane up the A34, but that has the disadvantage of not really going anywhere useful, unless you happen to live on the A34. (The A38 joins the University campus and two hospitals to the centre of town, and joins up with other off-road cycle routes.)
The lanes have priority at side-roads
[1], and some of the traffic lights give the cycleway priority (either green by default for cyclists, or stopping motor traffic and turning green in response to a cyclist approaching). This 'minor' detail is a step-change in approach to cycle infrastructure in the UK.
They're not perfect - there are some faffy junctions that slow you down
[2], and the changing sides (traffic light controlled with priority, see above) to fit available space adds complexity (but scores a political acceptance win of not reducing motor traffic capacity). There are snagging issues with drainage and use of tactile paving at bus stops, and they could do with being swept more frequently than they aren't. And the old adage of "Give a Brummie some tarmac, and they'll park a car on it" still applies. But they're a substantial improvement on the usual shared-use bollocks (a prime example of which can be seen where the blue lane peters out and connects to the infra that was built several years ago on the A38 Selly Oak bypass).
I reckon the A38 lane has been worth it, even if no cyclist ever used it, simply for the pedestrian improvements that were part of the re-working of the Bristol Road/Priory Road junction. Crossing the road there used to be terrifying.
Currently the battle is to stop the council from slipping back to their old ways (loss of priority, excessive multi-stage crossings) as they continue the A34 route. And hope that another pot of money from central government turns up to continue the longer-term plans.
FWIW, the 'Brimingham Cycle Revolution' has also surfaced the canals (with stupid chipseal-on-perfectly-good-tarmac, because that magically means they don't need to maintain it, but much better than the previous mudbath) within the city limits and I believe similar has been done from the Wolverhampton direction. IMHO the canals are a lovely day out, but they're not cycle infrastructure. They're completely unsuited for any real volume of traffic, physically inaccessible to many cyclists, social safety can be an issue, and they can be downright dangerous in winter.
[1] The drivers are learning, slowly. Helped by some no-nonsense policing. You need to exercise Brit-who's-just-got-off-the-ferry disbelief as you approach them.
[2] The crossing of the Middleway is less faffy than it looks - while technically two-stage, you get a green straight through for part of the cycle. The handling of the Edgbaston Park Road junction is tedious, and could do with an escape route onto the carriageway for those heading southbound and intending to turn left up ahead.