Single lady here, so not quite the same situation but there are some similarities.
You're unlikely to get any assistance through the NHS unless you also have a diagnosed condition which would require fertility treatment, or can evidence a lengthy history of trying to become pregnant and not succeeding. And even then, in many areas the assistance available might not amount to much. By all means have a chat with your GP to see if they can offer anything, but it's likely that any costs will mainly fall to you.
Since the main missing factor here is sperm, some single women and lesbian couples choose to enter into private arrangements with a male friend or someone else willing to donate. This option has very few upfront costs, but it's important to be aware of possible complications around parentage - this can vary by circumstance, so look into it. In addition some people have had bad experiences with dodgy people on donor sites, so again, proceed carefully if you choose this option.
If you decide to go via a clinic, the legal aspects are generally clearer and you have a lot more insight into what's going on medically. Usually, you'd have some initial tests done and then they'd make recommendations about whether IUI (intra-uterine insemination) or IVF would be your best option.
A couple of things you might want to consider: one thing some lesbian couples choose to do is to essentially do a partner-to-partner egg donation - so, one partner contributes the eggs, and the other carries the pregnancy, meaning both have a direct physical link to the child.
And, you might want to consider whether you'd be willing to donate any of your eggs (assuming you meet the criteria to do so) - this is obviously a big decision, but egg-share donors can access much cheaper IVF treatment which can make it more affordable, and you may enjoy the aspect of helping someone else. (I've been an egg-share recipient twice, and both times my donors were women in same-sex relationships. Sadly those cycles didn't work for me but I'm really pleased to have been able to help them - I know my second donor did become pregnant herself as a result of that treatment cycle).
Sperm is EXPENSIVE. Expect to pay about £1k a go for sperm.
Some people choose to go abroad for treatment, and this can be cheaper than the UK. Be aware, though, that a) donor anonymity laws in Europe are often different, meaning any resulting child would never be able to receive any information about their donor, and b) some of the destination countries for fertility treatment have laws restricting who can access said treatment - single women and same sex couples may not qualify.
Depending on your age, it might be a good idea to have some initial investigations done via a clinic sometime soon, just to establish whether there are any factors which might be a barrier to conceiving. But otherwise, maybe just start talking through what approach you guys might like to take, and work from there. (And, yeah, probably build up the savings some more. This stuff is pricey.)
Very best of luck for the future and best wishes for your wedding!