Unfortunately, it could affect your connections, at least temporarily around the installation phase; and possibly longer term through cross-talk.
Keep in mind that for the vast majority of phone connections in the UK, the actual physical cable, cabinets etc from the exchange to the house, are owned and maintained by "Openreach BT".
So for the majority, no matter the BB Provider (ISP), a Line Rental is paid in some manner to Openreach BT - and any line faults etc are dealt with by OR or its own sub-contractors - NOT your ISP, who has to report such problems to Upenareach or possibly BT Wholesale.
The cable from the PCP (typically grey or green distribution cabinet) has a cable running to your house, either overhead (poles etc) or underground (chambers about every 200 yards normally in the footpath), ending in your house at nrmally a Network Terminating Unit - ie the white outlet of the same size as a single electrical outlet.
The cable normally has four conductors/wires, of which two will be in present use for "your phone line/connection".
The other two are effectively stand-by and not connected at either end, in case a problem occurs with the working pair, either as a pair or as single wires.
There would be total loss to both users if that cable was broken by say, a tree falling over it (overhead) or an excavator breaking through it (underground), just as there is such risks with only one user at present.
So the simplest, quickest and CHEAPEST way to get a second BB connection technically is to make use of that second pair of wires - involving Openreach making the necessary connections at each end including back to the exchange via the PCP cabinet, additional NTE etc
You would automatically lose the stand-by aspect by not having that pair available.
It would also imply greater chances of cross-talk because of the intimate close proximity of another BB signal along the same length of cable.
The other additional ISP need NOT be BT or any of its subsidiaries - although that may lead to further complications!
Another way would be that the new BB ISP arranges that Openreach provide a separate cable - but that I understand is expensive.
Another possibility but also with more cost implications would be some form of satellite or line-of-sight radio arrangement.
As to whether any of those varieties are available in your area ...?