Council tax and bills about £600!? You are clearly bonkers if you think that
I assumed that was per month.
£200-300 council tax
£150 utilities
£50 water
£100 for home insurance, broadband, TV licence, subscriptions.
Seems about right?
If they have no housing costs, they can probably afford a reasonable life in London on £35k, especially if their income increases if the OP works longer hours or they get promotions.
Also people in London always say lots of free or cheap things to do and you don't have the running costs of a car, plus free travel for DC and tube passes for adults mean that they'll spend less on transport than a lot of people.
Plus as key workers (low paid NHS?) their incomes are probably secure so won't be out of work.
But I still don't think it's wise to commit all your wealth to your own property if that's what the OP is doing. The stamp duty on a property that amount is huge and just gone when they buy. Plus any work on the house is likely to beyond their reach as trades in the area will be expensive.
So probably more sensible to look around the £2M mark and keep a chunk back for major expenses, pensions, DC education, emergencies etc etc