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La Sportiva Tarantula and Tarantulace seem pretty popular as first climbing shoes, although I've not used them myself so can't comment on them. As you say, fit is the most important thing so ordering a few pairs—potentially in a range of sizes—is a good idea.
Go Outdoors is worth checking as they sometimes have heavily discounted shoes, although it can be a bit hit or miss: https://www.gooutdoors.co.uk/climbing/climbing-shoes/
Any suggestions for baby's first climbing shoes*? I know fit is very important so think I'll have to order a few pairs and keep the best ones as shops aren't likely to let me try things on these days. For mainly bouldering, always indoors.
* me, not an actual baby