Hi there - my husband Chris trains all ages including 50+, all abilities and is well inside your budget and offers 10% off block bookings of 6+ classes. He also trains two people for the same price as one so you could train with a partner if you prefer. I realise you are looking for a recommendation, but If you Google 'Generation Fit London' you will see his google reviews and you might also find older recommendations on this site.
He is based in SW18 and is mobile and provides all the equipment and would be happy to have a no obligation chat if you'd like to WhatsApp him 07887 897898 and have a trial session to see iff he is right for you
website is
www.generationfit.london
and Facebook and Instagram @generationfit.london