Hello and welcome to the Forum!
Whether you're a beginner or elite, you really need to be fitted properly and to this end I recommend Everglides, who are in Gosport on the south coast. Easily reachable from London by car or public transport (train then bus from Fareham). You make an appointment and they spend about 2 hours (often longer) measuring, checking your pressure points, arches, how you stand, walk, etc and helping you try on and select the boots and blades best suited to your feet and aims. They heat hold and punch out if necessary and you can go back to have things tweaked. I currently live in Cornwall but travel to them (a 5 hour journey each way) because I know and trust them to fit me properly. They have saved me a lot of money, time and pain!
Whilst London has rinks, I don't know of anywhere reliable in the capital to be fitted properly. Several skaters I know who used the London Skate Centre were "fitted" with completely unsuitable equipment (put into completely the wrong style of boot for their feet and level - nobody needs a low cut dance boot when they are a complete beginner!) and at least one of them actually gave up and now stands glumly watching his children skate. I have heard good things about Love Ice Skating in Chelmsford but I have no personal experience of their fitting expertise.
Edea, Jackson, and Risport are the most popular brands of boot in the UK - and, indeed, the world - but Riedell and Graf are sometimes available here, too. I like Graf but find Riedell very painful indeed. I dislike Edea for the loose fit around the ankles. Jackson and Risport are better for my feet. Avoid anything from SFR, Oxelo, LMNADE. If it's available at Decathlon then you don't want it!
Blade-wise, you may be fitted with a standard set (Edea Overture, Jackson Freestyle, Risport Electra but nothing lower in those ranges) but if you're fitted with a boot only and need to choose a separate blade, then Coronation Ace, MK Professional or Legacy 7 are your choices. Many people hesitate to recommend Legacy as it has a flatter spin rocker, but at your level it will be absolutely fine and should comfortably take you a long way in your skating journey. There are currently shortages of both boots and blades and Coronation Ace (pretty much always my first pick for almost any skater) are difficult to find in the more popular sizes. Everglides don't - or certainly didn't last time I was there - stock MK Professional (no idea why) so Legacy 7 would be the way to go. Do not be tempted by the "Lite" versions of Aces or Pros. I have yet to find anyone who is actually happy on them and they really aren't suitable for beginners at all.
Always happy to help, so don't hesitate to ask!