Low mood is either too little estrogen or the wrong sort of progesterone.
You need to give 3 pumps a few weeks and then consider changing from micronised progesterone to something else- eg Nalvee, Mirena coil, or even the mini pill in a larger dose.
You don't need any referral form your GP for those consultants for self-pay. They take direct bookings, although your insurers will need a referral. Please also check they accept insurance payments as some don't.
It's incredibly tricky giving feedback on consultants, as it's such a personal relationship between two people.
I've seen posts online (not about those doctors) where some have loved a consultant and other women have been disappointed. I think you just have to read their bios and take a chance. I'd not be happy making recommendations online as it can stray into criticisms which isn't fair, and I do know something of two of the drs you mention.
Most consultants have a waiting list of around a month.
Having said this, if all you need is perhaps an increase in estrogen, or a change in progesterone, which seems likely, any GP should be able to do that- it doesn't necessitate an NHS referral - they are usually for very complex cases after cancer or for women with high risk factors. This isn't a criticism of you but your GP!
On the other hand if you want to pay for time (30 mins or more) it's worth going privately. Do check though that your insurance will continue to fund HRT consultations if you need those - that is quite unusual. My insurers are great but won't cover ongoing consultations for something that can last for years.