Ok Lower - I'll have another go at it
Firstly - Liecestershire is way out of my patch and I fear we we would be OTT on costs, although your type of job is
standard to us - some one might be able to help you although from the details posted so far your well one the way so maybe you can complete it yourself.
Anyway - What I am about say will probably raise a few eyebrows and could even get me shot at dawn by a good many members of this forum, never being worried about such reactions, I'll stick my head up to see how many seconds it takes to start the onslaught.
We / I do a great number of restoration works on Victorian Floors - yours is not is Art-Deco period but the principle is the same, and we are often faced with
cement residues, not as much as yours I admit. We use acid mixtures - normally hydrochloric but occasionally sulphuric in low solution strengths. It is safe on unglazed quarry tiles, victorian tiles etc, however you need be careful when using it on old glazed tiles as it can etch the glaze.
I apply it as a low strength solution - max 15% either as brick acid / brick cleaner etc using an old stiff paint brush, rotary action. It will foam and hiss a bit as the lime in the
cement is broken down / dissolves and as the ph of the result residue reachs neutral it will cease to work. Clean this off and re-apply until the
cement is all lifted off.
Now the critical bit - double rinse the areas and then after all acid operations has been completed, take another clean cold bucket of water, add a small - I mean small !! quanity of caustic soda to the rinse water and wash the floor really well. Rinse with clean water again ! and dry out.
If you have dissolved any grout - so what, simiply regrout - bearing in mind that any repairs etc will need regrouting anyway.
We would use either LTP / HG impregnator - min 2 coats to protect the floor and restore the colour and then either LTP Satin Ironwax or our preferred choice HG Golvpolish satin - 5 OR 6 Coats and bob is your uncle.
I forgot to mention that we are helped by having some pretty effective machinery to aid our cleaning processes - too expensive for DIY guys but it our investment with a whole range of pads , brushes and diamond cutting gear too. You will not have this available to you, but you will be able to achieve comparable results - it will just take you a lot longer and more effort.
Just to show what can be done - Here's a typical example of our restoration work