If you intend to pursue a Canadian licence then organise a medical examination at least 8 weeks in advance of commencing training. If the exam is done in UK then you could arrange for a Part-MED exam to be conducted simultaneously. See database of Canadian examiners (
link). Transport also maintains a searchable database of flight training units (
link). Canada does not require foreign nationals undertaking flying training to undergo a security threat assessment. Foreign nationals do not require a study permit when pursuing courses of 6 or fewer months in length. Section 188 of the Immigration and Refugee Protection Regulations refers (
link). Canadian licences and ratings may be converted to US equivalents under the terms of a bilateral aviation safety agreement. See the implementation procedures (
link) or advisory circulars published by Transport and FAA (
link). A TSA security threat assessment will be conducted in the background but you do not need to apply for it.
To fly a UK-registered Part-21 aircraft or, as a UK resident any non-UK-registered aircraft in a non-commercial operation, in UK on and after exit day an appropriate licence granted converted or validated under the Aircrew Regulation will be required. Under UK General Exemption E 4863, ORS4 no. 1301 of 29 Mar 2019 (
link), a US airman certificate may be used as rudestuff has mentioned. No equivalent exemption exists for Canadian licences. See licence map applicable on exit day (pdf
link). That map takes account of amendments intended to be made to the UK Air Navigation Order by the Aviation Safety (Amendment etc.) (EU Exit) Regulations 2019 effective on exit day (
link).