Indeed, the train from BHX to Euston takes only 10-20 mins longer than the train or tube from LHR. Much more comfortable too. BHX is a horrible airport IMO but it's still a good choice if the price and schedule works. Do not ignore EDI as a possible entry point either.

Also bear in mind that if you have separate tickets India-EU and then EU-UK, you may be asked to demonstrate that you are allowed to enter the EU country / Schengen when you check in, irrespective of the checked luggage situation.