Unfortunately, in my experience Cheers is correct in saying that £250-300 per week is at the low end for a one bedroom, at least north of the river, which is generally more expensive than south. I’m sure you’ll be able to find a decent place for that amount but either you or your husband may end up with a commute longer than 30 minutes if you factor in walking or a bus ride to/from the tube.
Neighbourhoods along the Northern line would be good for both of your jobs, but it’s often very, very crowded.
North of the river, the Victoria line would also work. Warren Street is very close to Goodge Street (and your husband could either walk or change to the Northern line) and it’s an easy change to the Bank branch of the Northern line to London Bridge at Euston if you’re travelling south.
I’m more familiar with North London some suggestions are:
Tufnell Park, Archway, Upper Holloway, Highgate on the Northern line. Finsbury Park, Stroud Green, Crouch End (though this requires a short bus ride to Finsbury park tube). Highbury and Islington would be great too but it would be a real struggle to find something in your price range in my opinion.
I’m less familiar with South London but Clapham is great but I think you’d struggle to find a good 1 bedroom in your price range.
Good luck. If you need more advice or have questions about a specific area I’m sure you’d get more responses.