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Get a selection of pay as you go SIMs (or ask your neighbours) and do speed tests on your phone - see which one is fastest and then go with them.
I bought a Huawei B535 router for £90 and am currently on a £35 a month giffgaff sim. Speeds are 50-90 mb/s down and 10-20 upload. Service so far has been fairly reliable.
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We've been 4g EE for quite a while for the same reasons as you. Shitty 1930s wiring and BT wanting too much money to replace it.
Weve got strange circumstances as EE rent a spot on our roof for a mast so we have excellent signal. We get anything between 30Mbps and 200MBps depending on time of day. We've found it to be a very good service to get away from our shit broadband while we wait for hyperoptic to do their thing.
I dream of 20Mbps, we get 8 down and .25 up due to a 1930's block and paper wrapped solid copper phone lines.
Therefore I am seriously considering the 4G/5G option - looking at the coverage checkers EE seems to be the one that covers Forest Hill the most. Can anyone comment?