Afraid your description totally fits a semi blocked mains pipe and as VeniVidiVoxi has said someone (not pregnant you please - they're heavy and potential for bad things) needs to lift the manhole cover and watch while you flush the loo and empty some bath water. You should see it come straight through. (If in any doubt add cheap food coloring.)
If the water isn't coming straight through it needs rodding (from the manhole cover end)
Dont put drain unblocker down unless you know there are no plastic pipes involved!
If this gets worse or it warms up I'm afraid the next thing is the actual waste contents of the blocked sewer suddenly come back up through the toilet and bath and flood your home. (It's sudden when it happens, no bigger build up to warn you) What you have now is mainly the water used to flush your waste, not waste itself.
Many London properties have the old five inch mains pipes (with a rat trap in them to stop rats coming up from the sewer) and are insufficient for properties now divided into flats.
Be warned if the manhole is down the side of the house or equally close, check for broken bricks and render, air vents etc as IME they can bail out sewer contents onto the ground then jet the backed up waste straight into the void under your flat, so make sure any vents etc are covered, have the other half watch, and you may need many buckets of water, bleach and a stiff broom to clean up afterwards.
I'll love to be wrong but suffer serious sewage floods in my home on a regular basis thanks to my neighbours, (who deny being the wet wipes, tampax, and condom culprits allowing landlords to treat it as a 'neigbour dispute' not a plumbing matter.) and it's a miserable experience and proper cleaning if it gets to floorboards and skirting goes on for ever.
Immediately I'd advise removing anything you can from floor level including off bedroom floor if next to bathroom, don't wait for EH to deal with this, get the manhole cover lifted fast. If your financially solvent get a drain co out now and fight over who pays later, the misery it will save you is worth it if you can.